Multipath Routing Ph.D. Research Proposal Ron Banner Supervisor: Prof. Ariel Orda March 2004
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Multipath
Routing
PhD Research Proposal
Ron BannerSupervisor Prof Ariel Orda
March 2004
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
What is Multipath Routing
Multipath Routing is the method of establishing multiple paths between given source-destination nodes within the network
Advantages of Multipath Routing
Survivability
Provides redundancy
Congestion avoidance Improves network utilization
Provides load balancing
Management and control
Provides better performance in the presence of
selfishunregulated behavior
Previous Research
Survivability Mainly solutions that focus on the establishment of pairs of
disjoint paths (eg the 1+1 and 11 protection architectures)
Congestion avoidance Mainly heuristics (eg ECMP) Online no previous work for multipath routing
Management and control No previous work on the degradation of network performance due
to selfish behavior of users that employ multipath routing
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
What is Multipath Routing
Multipath Routing is the method of establishing multiple paths between given source-destination nodes within the network
Advantages of Multipath Routing
Survivability
Provides redundancy
Congestion avoidance Improves network utilization
Provides load balancing
Management and control
Provides better performance in the presence of
selfishunregulated behavior
Previous Research
Survivability Mainly solutions that focus on the establishment of pairs of
disjoint paths (eg the 1+1 and 11 protection architectures)
Congestion avoidance Mainly heuristics (eg ECMP) Online no previous work for multipath routing
Management and control No previous work on the degradation of network performance due
to selfish behavior of users that employ multipath routing
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
What is Multipath Routing
Multipath Routing is the method of establishing multiple paths between given source-destination nodes within the network
Advantages of Multipath Routing
Survivability
Provides redundancy
Congestion avoidance Improves network utilization
Provides load balancing
Management and control
Provides better performance in the presence of
selfishunregulated behavior
Previous Research
Survivability Mainly solutions that focus on the establishment of pairs of
disjoint paths (eg the 1+1 and 11 protection architectures)
Congestion avoidance Mainly heuristics (eg ECMP) Online no previous work for multipath routing
Management and control No previous work on the degradation of network performance due
to selfish behavior of users that employ multipath routing
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Advantages of Multipath Routing
Survivability
Provides redundancy
Congestion avoidance Improves network utilization
Provides load balancing
Management and control
Provides better performance in the presence of
selfishunregulated behavior
Previous Research
Survivability Mainly solutions that focus on the establishment of pairs of
disjoint paths (eg the 1+1 and 11 protection architectures)
Congestion avoidance Mainly heuristics (eg ECMP) Online no previous work for multipath routing
Management and control No previous work on the degradation of network performance due
to selfish behavior of users that employ multipath routing
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Previous Research
Survivability Mainly solutions that focus on the establishment of pairs of
disjoint paths (eg the 1+1 and 11 protection architectures)
Congestion avoidance Mainly heuristics (eg ECMP) Online no previous work for multipath routing
Management and control No previous work on the degradation of network performance due
to selfish behavior of users that employ multipath routing
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Notations
G (VE) ndash Directed GraphV - Collection of nodesE ndash Collection of links (edges)
P(st) -Collection of all paths from s to t(st) ndashflow demand from s to tde-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
ee p
D p dD(p) ndash the end-to-end delay of path p ie
C(p) ndash the capacity of path p ie (p) ndash the reliability of path p ie
min ee pC p c
1 ee E
p p
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Summary of results Survivability
We provide a quantitative framework that specifies the desired level of survivability against single failures
c=20 p=005
c=30p=005
c=30 p=005
c=30
p=0
05
c=10 p=005c=30 p=0
c=30 p=005
S T
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Summary of results Survivability
We developed optimal polynomial schemes for 11 and 1+1 protection that consider important tradeoffs Survivability vs bandwidth Survivability vs feasibility hellip
No need to establish connections that consist of more than two paths
Derived a new ldquohybridrdquo protection architecture that has several advantages over both the 11 and 1+1 protection architecture
Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Summary of resultsCongestion minimization-offline
Goal Minimize network congestion when all demands are known in advance
Cope with constraints Delay jitter End-to-end delay Number of paths
Minimizing the congestion under end-to-end delay andor delay jitter NP-hard Pseudo polynomial solution optimal approximation scheme
Minimizing the congestion while restricting the number of routing paths NP-hard 2-approximation scheme
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Summary of results Congestion minimization-online
Goal Minimizing the network congestion when demands arrive one at a time
Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio
Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization
Our algorithm is best possible
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Summary of resultsSelfish multipath routing
Goal Investigating the degradation in network performance due to selfish behavior of users
Given a load-dependent performance function qe(fe) for each link we consider bottleneck network objectives ie MaxeEqe(fe) and additive network objectives ie
Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements
e ee E
q f
infin1
infinM Additive
Bottleneck
Network objective
Routing approach Multipath
RoutingSingle-path
Routing
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The tunable survivability concept
Current survivability schemes typically offer two degrees of protection against single failures Full (100) protection No protection at all
In practice the requirement of full protection is often too restrictive In many cases it is infeasible (N Taft-Plotkin B Bellur and R Ogier)
In other cases it is very limiting (G Maier A Pattavina S De Patre and M Martinelli)
Tunable survivability enables to consider valuable tradeoffs Survivability vs bandwidth Survivability vs feasibility Survivability vs end-to-end delay hellip
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Survivable connections
p-survivable connection a collection of paths (p1p2hellip pk)P(st)timesP(st) timeshelliptimes P(st) that upon a link failure has a probability of at least p that at least one path out of (p1p2hellip pk) remains operational
The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum Bge0 such that nmiddotBlece for each link e that is common to n paths from (p1p2hellip pk)
The probability of a survivable connection to remain operational upon
a single failure is the probability that all the common links are
operational upon that failure ie 1 2
1- k
ee p p p
p
The bandwidth of a survivable connection with respect to the 11 protection
architecture is the maximum Bge0 such that Blece for each e that belongs to a
path in (p1p2hellip pk) It is also
1 2
min ke p p p
ec
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof (sketch for the 11 protection) We shall construct only from the links that belong to paths in
(p1p2hellip pk) Therefore the bandwidth of is at least that of (p1p2hellip pk)
Formal proof
1 2 st stp p P times P
1 2p p
1 2p p
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough we focus on survivable connection that consist of two paths
The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem
The flow demand is set to 2∙B flow units
A link in the original network
Links in the transformed network
Discard the link Ce
ltB
BleCelt2∙B
Cege2∙B
ce=B we=0
ce=B we=0
ce=B we=-ln(1-pe)
cepe
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min cost flow is B-integral
The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2middotB flow units) into a flow over two paths p1 p2 such that f(p1)=f(p2)=B
Since the flow has a minimum cost has a minimum value
Therefore (p1p2 ) is a connection with a bandwidth of at least B that maximizes hence it maximizes
1 1
ln 1e e ee E e p p
f w B p
1 1 1 1
ln 1 ln 1 e ee p p e p p
p p
1 2
1 ee p p
p
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the maximum probability to remain operational upon a failure It is also the most survivable connection with a bandwidth of at least B=0
The widest p-survivable connection is the p-survivable connection with the maximum bandwidth
How to establish the widest p-survivable connection
Idea search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection
It is enough to perform a binary search over the set Why
The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm Why
12 ec e E kk
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The only difference in the reduction lies for the links that have capacities in the range [B2B]
For 11 protection only one of the paths carries B flow units
Hence all links that have a capacity in the range [B2B] can concurrently be employed by both paths
A link in the original networkLinks in the transformed network
Discard the link CeltB
CegeB ce=B we=0
ce=B we=-ln(1-pe)
cepe
Establishing Survivable Connections for 11 protection
Go to 1+1 reduction
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The tunable survivability concept gives rise to a third protection architecture
Reduces the congestion of all links that are shared by both paths wrt 1+1 protection
Upon a link has a faster restoration wrt 11 protection Provides the fastest propagation of data However requires additional nodal capabilities
The Hybrid protection architecture
S T
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The hybrid architecture transfers through each link exactly one duplicate of the original traffic
Hence the bandwidth of (p1p2) with respect to hybrid protection is
Hence by definition all schemes for 11 protection apply for hybrid protection
The Hybrid protection architecture
Go to Def
1 2
min e p p
ec
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Simulation results
We quantify how much we gain by employing tunable survivability instead of full survivability
Random networks 10000 Waxman topologies 10000 Power-law topologies Explain the construction
08
1
12
14
16
18
2
22
24
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1
1)
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Simulation results
08
1
12
14
16
95 96 97 98 99 100
level of survivability p
Power-Law Waxman
Ban
dwid
th r
atio
(1+
1)
1
12
14
16
18
2
22
24
26
28
3
95 96 97 98 99 100
degree of survivability pPower-Law Waxman
Fea
sibi
lity
rat
io
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Agenda
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Problem formulation
Goals Minimize network congestion when all demands are known
in advance Cope with constraints (delay-jitter delay number of
paths)
Performance Objective network congestion factor
Minimizing
RFC 2702 and others
No link becomes over-utilized
More room for future traffic growth by maximizing the
common scaling factor
max e
e Ee
f
c
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Requirements for practical deployment
Restricting the delay-jitter among all routing paths RFC 2991 Avoid the ldquofast retransmitrdquo mode Reduce buffering requirements
Limiting the number of paths per destination S Nelakuditi and Zhi-Li Zhang Reduce the tendency of packet reordering Reduce overhead Simplify the schemes that distribute traffic
Bounding the end-to-end delay of each path
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Computational Intractability
Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard Proof
Minimizing the network congestion factor under the delay jitter restriction is NP- hard Proof
Minimizing the network congestion factor under the restriction on the number of paths is NP-hard Proof
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing congestion while restricting the number of paths
Observation The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths
Proof Let f be a path flow that has the smallest network congestion factor α among all path flows that transfers flow units from S to T over at most K
paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2 flow units from S to T over at most K paths
Round down the flow f(p) over each path to a multiple of K Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
Since f transfer 2 flow units over at most K paths fR transfers at least
flow units from S to T
fR is a K - integral path flow that transfers at least flow units from S to T and has a network congestion
factor of at most 2∙ α
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under integrality restrictions
A K-integral path flow admits at most K paths
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
The network congestion factor of all K-integral path flows belong to
The flow over each link is integral in K and is at most Hence for each eE it holds that
In particular
0e
i e E i KK c
0 e
e e
fi i K
c K c
max 0 e
e Ee e
fi e E i K
c K c
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under integrality restrictions
Goal Find a K-integral path flow that has the minimum network
congestion factor in
Solution
Find a path flow with the smallest such that
the following procedure succeeds
multiply all link capacities by a factor of α
Round down the capacity of each link to a multiply of K Since the flow must be K-integral such a rounding has no affect
Apply a maximum flow algorithm that returns a K-integral link flow
when all capacities are integral in K
If the link flow transfers flow units from S to T return Success
Else return Fail
0 e
i e E i KK c
0e
i e E i KK c
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under end-to-end delay restrictions - linear program
It is straight forward to extend the linear program to the multi-commodity case
The path flow is constructed using a variant of the flow decomposition algorithm
The complexity incurred by solving the linear program is polynomial in D
The number of variables is O(MD)
The number of constraints is O(MD)
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Approximation Scheme
Goal reduce the value of the end-to-end delay restriction D Delete from the network all the links with a delay degtD Delay scaling
Apply the linear program for the new instance As the new instance relax the original instance the congestion is
not worse then the optimum Convert each non-simple path into a simple path Total error for a path N New end-to-end delay D+ N=D∙(1+є)
D D D= where e
e
dd
N
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under delay-jitter restrictions
Idea restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths
It is sufficient to add the linear program a minimum end-to-end delay restriction L New Linear Program
Given a delay-jitter restriction J and an end-to-end delay D For each L[0D-J] solve the new linear program with a minimum
and a maximum end-to-end delay restrictions L L+J respectively
Scaling down the end-to-end delay restriction D produces an є-optimal approximation scheme for the case where dmax=O(J) Details
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Selfish Routing
Network users are selfish Do not care about social welfare Want to optimize their performance
A central Question how much does the network performance suffer from the lack of global regulation
A flow is at Nash Equilibrium if no user can improve its performance May not exist May not be unique
The price of anarchy The worst case ratio between the performance of a Nash equilibrium and the optimal performance
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Previous Work
[KoutsoupiasPapadimitriou] First paper to propose quantifying the cost of lack of
regulation Concentrated on two node networks
[Roughgarden] General networks Infinite number of users users route traffic along the minimum latency path The price of anarchy is unbounded
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Model
A set of users U For each user a positive flow demand u and a
source-destination pair (sutu)
For each link e a performance function qe(∙) qe(∙) is continuous and increasing for all links
Users behavior Users are selfish They optimize bottleneck objectives
Network Bottleneck objective Additive objective
e ee E
C f q f
e ee E
B f Max q f
0
( ) ue
u e ee E f
b f Max q f
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Non-uniqueness of Nash Equilibrium
s t
One user wants to transfer 1 unit from s to t Assume that qe(fe)=fe for each eE
(fp1=1 fp2=0) amp (fp1=0 fp2=1) are Nash flows with respect to unsplittable flow vectors
(fp1=05 fp2=05) amp (fp1=025 fp2=075) are Nash flows with respect to splittable flow vectors
We identified two different Nash flow for each routing approach
e2
e1
e3
p1
p2
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Existence of Nash Equilibrium
Definition integral flow vector is a feasible flow vector where is integral in for each user u U and pP
Theorem Considering integral flow vector there exists a Nash equilibrium for each N+ The existence of NEP for Single-path Routing corresponds
to the case where N=1 The existence of NEP for Multipath Routing corresponds to
the case where Nrarrinfin However still needs to prove for the case where ldquoN=infinrdquo
The proof of the theorem
1
N
u
N
1
N
upf
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often unbounded Roughgarden the price of anarchy is unbounded Papadimitriou the price of anarchy is
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed then the price of anarchy is 1 Proof
Braess paradox the addition of links to noncooperative networks can negatively impact performance of all users However cannot occur for multipath routing (when qe(0)=0)
log
log log log
M
M
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Price of anarchy is at most M with additive objectives
Theorem Given an instance [G(VE) Uqe()] If multipath
routing is allowed than the price of anarchy with respect to additive network objectives is M
Proof Let f and f denote a Nash and an optimal flow correspondingly
Therefore B(f)leB(f)
Therefore maxeE qe(f) lemaxeE qe(f)
Hence sumeE qe(f)le M∙maxEqe(f) leM∙maxeE qe(f) leM∙sumeE qe(f)
Corollary Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing Additive network objectives Bottleneck network objectives
4
3 2e e
2
3 ef
e eq f e
1
2 ef
e eq f e
A=
B= 2∙
S T
Additive
Bottleneck
Optimal flow
Nashflow
4
3e
2
3e e
e
Price of anarchy
3e
43 2
23
e e
e e
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The Model
Requests arrive one at a time and there is no a priori knowledge regarding future demands
Each request specifies the source sr and destination tr
the requested flow demand r
the maximum number of routing paths kr that can carry the demand
Goal Route all demands while minimizing the network congestion factor
For the case were demands are limited to single an O(logN)-competitive strategy was derived by Aspnes Azar Fiat Plotkin Waarts
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input sequence
The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm
In our case the performance is the network congestion factor
The entire requests sequence is denoted by R
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under integrality restrictions
A path flow is K-integral if the flow of each request rR over each path is integral in rKr
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
Proof A K-integral path flow employs at most Kr paths for each rR
Corollary minimizing the congestion while restricting the flow to be integral in K is a 2-approximation scheme
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Online solution
Upon the arrival of the nth request Split the request to Kn successive requests to transfer nKn flow
units
Employ the online strategy of plotkin at el to route the demands over single paths
Plotkinrsquos online strategy produces a competitive ratio of O(logN)
Therefore we establish an online strategy with a competitive ratio of O(logN) for K-integral path flows
Therefore we establish an online strategy for our original problem with a competitive ratio of 2O(logN)=O(logN)
sn
nKn
nKn
nKn
tn
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
A Lower Bound of Ω(logN) for Multipath Routing
S
VN
VN-1
V3
V2
V1
M 11T
N
O
21T
22T
31T
32T
33T
34T
log 2
NN
T
log 1NT
log 2NT
M
The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K
2K
N
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
A Lower Bound of Ω(logN) for Multipath Routing (cont)
After logN requests the network congestion factor is at least frac12∙logN
The optimal offline algorithm can achieve a network congestion factor of 1
O
S
VN
VN-1
V3
V2
V1
M 11T
N21T
22T
31T
32T
33T
34T
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
A Lower Bound of Ω(logN) for Multipath Routing (cont)
There exists a lower bound of frac12∙logN for networks with at most Nrsquo=N∙logN+Nle2N∙logN nodes
We have to show that frac12∙logN=Ω(logNrsquo) Indeed there exists Cgt0 and NgtN0 such that
logNrsquo=logN+log(2middotlogN)=logN+log2+loglogN le C∙ frac12∙logN
There exists a lower bound of Ω(logN) for the best possible competitive ratio
Our online algorithm is best possible
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Agenda
Introduction amp summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Deepening the Current Work
Consider for the proposed schemes Distributed implementation Heuristic schemes with low complexity Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end delay
Establish a unifying scheme that bounds the number of paths the end to end delay of each path and the delay-jitter among all paths Online computation Offline computation
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Selfishness in Multipath Routing
In networks that have many users the price of anarchy with respect to additive metrics may be very large
If all users route their traffic with respect to bottleneck objectives the price of anarchy with respect to additive network objectives is at most M
Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M
Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics In that case what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Online Multipath Routing for finite holding time connections
We have established an online strategy for permanent connections (ie connections with infinite holding times) In practice the holding times are usually finite
There are online routing schemes with provable performance guarantees for the finite holding time case The holding time may be specified upon arrival Only the distribution on the holding time is known No information on the holding time
Investigate multipath routing for the finite holding time model Investigate the lower bound Establish corresponding multipath routing schemes
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Other Congestion Criteria
Thus far we measured congestion according to the most utilized links in the network
Although these links are the most severely affected by congestion other links are affected as well
Moreover there are cases where congestion is better modeled through non-linear optimization functions
Consider other optimization functions for congestion More general link congestion functions
Already considered in the work on selfish routing Congestion functions that consider all the links in the network
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Multipath Routing and Security
Only the target sees the whole data stream when it is split among several node-disjoint paths
Reconstructing the data stream is possible only at the target node
It is essential to Identify several node disjoint paths Assign a limited portion of the traffic over each path
Develop multipath routing schemes that engage this inherent advantage The solution must consider the requirements of multipath
routing
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a failure
Upon a path failure the data stream that traveled over the failed path may be split along the remaining paths Avoid additional path computation and resource reservation
Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path
Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Multipath Routing and Wireless networks
Energy Efficient Routing In wireless networks nodes have a limited power resources
(batteries) Energy consumption is proportional to node transmission rates Therefore splitting the traffic among several paths can prolong
the time until the first battery is exhausted Establish schemes that maximizes the networkrsquos lifetime while
considering the requirements of multipath routing
Survivability in wireless networks Standard survivability schemes establish pairs of disjoint paths If two links that belong to different paths are too near noise can
affect both links Establish schemes that consider the minimum physical distance
between two links that belong to different paths
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Fairness in Multipath Routing
A commodity may attempt to establish too many paths In order to maximize its bandwidth In order to maximize its survivability
This may come at the expense of other commodities Eg a commodity may use too many entries in a (limited)
routing table
Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in time
Often flow demands are not constant Users sendreceive data for short periods of time The TCP congestion control mechanism changes
transmission rates with time
Extend our model to cases where rarr (t)
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The End
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Two Paths are Enough
Theorem Let (p1p2hellip pk)P(st)timesP(st) timeshelliptimesP(st) be a p-survivable connection There exists a p-survivable connection that has at least the bandwidth of (p1p2hellip pk) with respect to the 11 (alternatively 1+1) protection architecture
Proof Remove from the network all the links that are not used by the paths of
(p1p2hellip pk) We have to show that there exists a pair of paths in the resulting network such that
Assign to each link two units of capacity and assign to all other links one unit of capacity
There exists a pair of paths that intersect only on links
from iff it is possible to define an integral link flow that transfers
two flow units from s to t
Hence it is sufficient to show that it is possible to define an integral link
flow that transfers two flow units from s to t
1 2 st stp p P times P
1 2 st stp p P times P
k
ii=1
e p
1 2 st stp p P times P
k
ii=1
p
1 2 k
i
i=1
p p p
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Two Paths are Enough
Proof (cont) However since all capacities are integral the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted Hence we left to show that it is possible to transfer two flow units from s to t
Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t
Hence according to the max-flow min cut theorem there exists a cut (ST) with sS and tT such that
Therefore since the capacity of all links is integral it follows that C(ST)le1
Hence since each link has at least one unit of capacity it follows that at most one link crosses (ST)
Denote this link by e Since C(ST)le1 it follows that cele1
Obviously all paths from (p1p2hellip pk) must traverse through e Hence Therefore by construction ce=2 which contradicts the fact that cele1
x y
x Sy T
C ST c lt 2
k
ii=1
e p
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set
If one path admits a link e then the bandwidth of the connection is at most ce
If both paths admit a link e then the bandwidth of the connection is at most ce2
Hence by definition there exists at least one tight link eE such that the bandwidth of the connection is either ce or ce2
Why O(logN) executions of a min cost flow algorithm The set contains 2middotM elements A binary search over the set enables to consider O(log2middotM)=O(logN)
values
12 ec e E kk
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The end-to-end delay restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p transfers a positive amount of flow then D(p)leD
The partition problem Given an ordered set of elements a1 a2 hellip a2n that constitute a set A with a size s(a)+ for each a A is there a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen such that sumaArsquo s(a)=sumaAArsquo s(a)
All link capacities are 1 Claim It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than frac12sumaA s(a) iff there is a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for 1leilen and sum
aArsquo s(a)=sum
aAArsquo s(a)
S(a1) S(a3) S(a5) S(a2n-1)
S T
S(a2) S(a4) S(a6) S(a2n)
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The end-to-end delay restriction is intractable
lt=lt= There is a a subset ArsquoA such that Arsquo contains exactly one element of a2i-1 a2i for
1leilen and sumaArsquo
s(a)=sumaAArsquo
s(a) The selection of the links that correspond to the elements of Arsquo and the zero
delay links that connect these links constitutes a path p Path p is disjoint to the path that the complement subset AArsquo defines Since all capacities are equal to 1 we have two disjoint paths that can transfer
together 2 units of flow The end-to-end delay of each path is frac12sumaA s(a)
=gt=gt There is a path flow that transfers two flow units over paths that are not larger
than frac12sumaA s(a) Let p be a path that carries a positive flow by construction p contains exactly
one element of a2i-1 a2i for 1leilen Since all the links have one unit of capacity p can transfer at most 1 flow unit Therefore there exists a path prsquo that is disjoint to p that transfers a positive
flow by construction prsquo=Ap Hence D(p) lefrac12sumaA s(a) and D(prsquo) lefrac12sumaA s(a) Therefore since D(p)+ D(prsquo)=sumaA s(a) it follows that sum
ap s(a)=sumaprsquo
s(a)=frac12sumaA
s(a)
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The delay jitter restriction is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t such that if path p1 p2 transfers a positive amount of flow then D(p1)-D(p2)leJ
Reduction from the problem with end-to-end delay restriction
S
T
A link with a capacity sumce and a zero
delay
It is possible to transfer flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer +sumce flow units in network B over paths
with delay jitter restriction W
S
T
A B
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
The restriction on the number of paths is intractable
A special case of our problem Is there a path flow that transfers flow units from s to t over at most K paths
The single source unsplittable flow problem Given a network G with a source s targets t1 t2 hellip tk and corresponding demands D1 D2 hellip Dk is there an assignment of traffic to paths such that for each 1leilek demand Di is routed over a single path without violating the capacity constraints
Claim There exists a path flow that transfers = D1+ D2 +hellip+ Dk flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D1 D2 hellip Dk to paths such that Di 1leilek is routed over a single path without violating the capacity constraints There is exactly one path from S to ti for each 1leilek Hence there are exactly K paths from S to T
that carry a positive flows There is at least one path from S to ti for each 1leilek However since there are at most K paths
there is exactly one path from S to ti for each 1leilek
S
t1 t2 tk
TD1
D2 Dk
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Waxman and Power-law topologies
Waxman networks Source and destination are located at the diagonally opposite
corner of a square area of unit dimension 198 nodes are uniformly spread over the square A link between two nodes uv exists with a probability which
depends on the distance between them δ(uv)
where α=18 β=005 Power-law networks
We assigned a number of out-degree credits to each node using the power-law distribution β∙x-α where α=075 and β=005
Then we connected the nodes so that every node obtained the assigned out-degree
exp
2
u vp u v
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing the congestion under delay-jitter restrictions
( ) ( )
0 0ede e
e O v e I v
f f v V s t D
DD D
( ) ( )
0 1ede e
e O s e I s
f f D
DD D
0
( )e
e O s
f
Minimize
s t
0
D
e ef c
D
De E
0ef D
0
0ef D
0 ee E D d D
0e E D D
( ) ( )
ede e
e I t e O tL D L D
f f
D D
D D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1leHleN-1 on the hop count Important in order to cope with routing loops
We present an approximation scheme for the case where dmax=O(J)
The number of variables is in the order of M∙H∙min DH∙dmaxle M∙H2dmax
The delay of each link is reduced to smaller integral value
Total error in the evaluation of the delay of each path is H∙Δ A pair of paths that originally have a delay jitter J may now
have a delay jitter J+H∙Δ Therefore in order to relax the new instance the delay jitter
restriction is
D D= where
2e
e
d Jd
N
JJ= H
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Approximation scheme for the restriction on the delay jitter
Assume that p1 p2 transfers a positive flow in the output We will show that D(p1)-D(p2)leJ(1+є)
deg deg
deg deg deg deg
1 2 1 2
1 2 1 2
1 2
1 2
1 1
1 1
J1 1
e ee e
e p e p e p e p
e ee e
e p e p e p e p
e ee p e p
d dD p D p d d
d dd d
d d p J p J H
JH N H
1
2 1 2
N
JJ N H J N J
N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output We will show that D(p) leD(1+є)
deg
deg
1
12
1 2
e ee p e p e p e pe e
d dD p d d p
D JD H N D N D N
ND
D N DN
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Existence of Nash Equilibrium
The joint strategy space is finite Each user selects at most N out of |P(st)| possible paths There are at most |U| users
By the way of contradiction assume that there is no Nash equilibrium Each profile in the joint strategy space has a player that can improve its
bottleneck Let ltf1f2hellip gt be a sequence of profiles such that for each two profiles
fi fi+1ltf1f2hellip gt exactly one user in fi+1 reroutes its traffic and improves its bottleneck with respect to fi
After a finite number of transitions between successive profiles we must encounter the same profile
Let u be a user that achieves the worst (not constant) bottleneck in all profiles ltf1f2hellipfn gt Let fk be the profile where u achieves for the first time the worst bottleneck
There exists in profile fk-1 exactly one user ursquo that improves its bottleneck
However since ursquo ships traffic through the bottleneck of u in fk ursquo is not improving its bottleneck
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
No price of anarchy for bottleneck network objectives
Theorem Given an instance [G(VE) Uqe()] If multipath routing is
allowed than the price of anarchy is 1proof Notations
f- Nash flow (f)- The collection of users that ship traffic through a network
bottleneck in f g- Path flow f without the users U(f) and their respective flows Ersquo ndash The collection of all network bottlenecks with respect to g P(e)- The collection of all paths that traverse through link e
Lemma g is a Nash flow that satisfies B(f)=B(g) bu(g)=B(g) for each user u(f) Proof
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
No price of anarchy for bottleneck network objectives (cont)
By contradiction assume the existence of a flow vector h B(h)ltB(g)
Since g is a Nash flow every path pP(sutu) where u(f) must traverse through at least one network bottleneck from Ersquo
Therefore for each bottleneck u(f)
Therefore
Therefore since the total traffic of every feasible flow vector that
traverses through the paths equals to the total
traffic that traverse through equals to both in g and
in h
u us t
u f e E
P P e
u us t
u f
P
e E
P e
u
u f
u
u f
u us t
e E
P P e
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
No price of anarchy for bottleneck network objectives (cont)
Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for each eErsquo
Therefore helt ge for each eErsquo Therefore the traffic that traverses through P(e) is smaller in h
than in g for each eErsquo Therefore the traffic that traverses through is smaller in
h than in g However this contradicts the fact that the total traffic of the
paths in is the same in flow vector h and g
Since B(g) is optimal and since B(f)=B(g) it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in (f) can only be worse than the bottleneck of g)
e E
P e
e E
P e
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Proof of the Lemma
Let Ersquorsquo be the collection of all bottlenecks with respect to f B(f)=B(g)
By definition the traffic that is carried over Ersquorsquo belongs only to (f)
Therefore since for each u(f) and pP it holds that for each eErsquorsquo
Therefore B(f)=B(g)
bu(g)=B(g) for each user u(f) Consider a user u(f) u must ship traffic through at least one link from Ersquorsquo in flow vector
f Since for each u(f) and pP it follows that u must also
ship positive traffic through a link from Ersquorsquo in flow vector g Since qe(ge)=qe(fe)=B(f) for each e Ersquorsquo it follows that bu(g)=B(g)
g is at Nash equilibrium Since f is a Nash flow every path pP(sutu) where u(f) must
traverse through at least one network bottleneck from Ersquorsquo
u up pf g
e ef g
u up pf g
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g Therefore every path p P(sutu) where u(f) traverses through one
network bottleneck with respect to g By contradiction assume there exists a user u(f) in g that can
improve its bottleneck
Let E(sutu) be the collection of all network bottlenecks in g on paths from P(sutu)
Let P(e) be the collection of all paths that traverse through e
u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link eE(sutu)
Therefore it must ship traffic to other paths from P(sutu) However we have shown that all other paths already traverse
through at least one bottleneck from E(sutu)
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Minimizing congestion while restricting the number of paths
Theorem The optimal network congestion factor of a K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most Kr paths for each request rR
ProofLet f be a path flow that has the
smallest network congestion factor α among all path flows that transfers for each rR r flow units from Sr to Tr over
at most Kr paths
f=2∙f is a path flow with a network congestion factor 2∙α that transfers
2r flow units from Sr to Tr over at most Kr paths for each rR
For each rR round down the flow f(p) over each path pP(srtr) to a multiple of rKr Let fR be the
resulting path flow
Given a network G(VE) and a
source-destination pair
For each rR f transfers 2r flow units over at most Kr paths Therefore fR
transfers at least r flow units from Sr to Tr for each rR
fR is a K - integral path flow that transfers at least r flow units from Sr to Tr for each rR and has a network congestion factor of at most 2∙ α
- Multipath Routing
- Agenda
- What is Multipath Routing
- Advantages of Multipath Routing
- Previous Research
- Notations
- Summary of results Survivability
- Slide 8
- Summary of results Congestion minimization-offline
- Summary of results Congestion minimization-online
- Summary of results Selfish multipath routing
- Slide 12
- The tunable survivability concept
- Survivable connections
- Two Paths are Enough
- Most Survivable Connections with a Bandwidth of at Least B
- Slide 17
- Establishing Most and Widest p-survivable Connections
- Establishing Survivable Connections for 11 protection
- The Hybrid protection architecture
- Slide 21
- Simulation results
- Slide 23
- Slide 24
- Problem formulation
- Requirements for practical deployment
- Computational Intractability
- Minimizing congestion while restricting the number of paths
- Minimizing the congestion under integrality restrictions
- Slide 30
- Minimizing the congestion under end-to-end delay restrictions - linear program
- Approximation Scheme
- Minimizing the congestion under delay-jitter restrictions
- Slide 34
- Selfish Routing
- Previous Work
- Model
- Non-uniqueness of Nash Equilibrium
- Existence of Nash Equilibrium
- No price of anarchy for bottleneck network objectives
- Price of anarchy is at most M with additive objectives
- Bad news for single-path-routing
- Slide 43
- The Model
- Evaluating the Quality of Online Algorithms
- Slide 46
- Online solution
- A Lower Bound of Ω(logN) for Multipath Routing
- A Lower Bound of Ω(logN) for Multipath Routing (cont)
- Slide 50
- Slide 51
- Future research
- Deepening the Current Work
- Selfishness in Multipath Routing
- Online Multipath Routing for finite holding time connections
- Other Congestion Criteria
- Multipath Routing and Security
- Recovery Schemes for Multipath Routing
- Multipath Routing and Wireless networks
- Fairness in Multipath Routing
- Time Dependent Flow Demands in Multipath Routing
- The End
- Slide 63
- Slide 64
- Establishing the widest p-survivable connection
- The end-to-end delay restriction is intractable
- Slide 67
- The delay jitter restriction is intractable
- The restriction on the number of paths is intractable
- Waxman and Power-law topologies
- Slide 71
- Approximation scheme for the restriction on the delay jitter
- Slide 73
- Slide 74
- Slide 75
- Slide 76
- No price of anarchy for bottleneck network objectives (cont)
- Slide 78
- Proof of the Lemma
- Slide 80
- Slide 81
-
Related Documents
