Aug 22, 2002Sigcomm 2002 Replication Strategies in Unstructured Peer-to-Peer Networks Edith Cohen AT&T Labs-research Scott Shenker ICIR.
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Aug 22, 2002 Sigcomm 2002
Replication Strategies in Unstructured Peer-to-Peer
Networks
Edith CohenAT&T Labs-research
Scott ShenkerICIR
Aug 22, 2002 Sigcomm 2002
Peer-to-peer Networks• Peers are connected by an
overlay network.• Users cooperate to share
files (e.g., music, videos, etc.)
Aug 22, 2002 Sigcomm 2002
(Search in) Basic P2P Architectures
• Centralized: central directory server. (Napster) Supports versatile queries, scope, legal troubles
• Decentralized: search is performed by probing peers– Structured (DHTs): (Freenet, Can, Chord,…)
location is coupled with topology - search is routed by the query. Scope, Only exact-match queries, tightly controlled overlay.
– Unstructured: (Gnutella, FastTrack); search is “blind” - probed peers are unrelated to query. Resilient to transient peers; versatile queries; Harsh scope/scalability tradeoff.
Aug 22, 2002 Sigcomm 2002
(replication in) P2P architectures
• No proactive replication (Gnutella)– Hosts store and serve only what they
requested– A copy can be found only by probing a host
with a copy
• Proactive replication of “keys” (= meta data + pointer) for search efficiency (FastTrack, DHTs)
• Proactive replication of “copies” – for search and download efficiency, anonymity. (Freenet)
Aug 22, 2002 Sigcomm 2002
Question: how to use replication to improve search efficiency in unstructured networks with a proactive replication mechanism ?
Aug 22, 2002 Sigcomm 2002
Search and replication model
• Search: probe hosts, uniformly at random, until the query is satisfied (or the search max size is exceeded)
Goal: minimize average search size (number of probes till query is satisfied)
• Replication: Each host can store up to copies (or keys=metadata+pointer) of items.
Unstructured networks with replication of keys or copies. Peers probed (in the search and replication process) are unrelated to query/item - Probe success likelihood can not be better, on average, than random probes.
Aug 22, 2002 Sigcomm 2002
What is the search size of a query ?• Insoluble queries: maximum search size• Soluble queries: number of probes until
answer is found. We look at the Expected Search Size (ESS) of
each item. The ESS is inversely proportional to the fraction of peers with a copy of the item.
Search size• Query is soluble if there are sufficiently many
copies of the item.• Query is insoluble if item is rare or non
existent.
Aug 22, 2002 Sigcomm 2002
Search Example
2 probes 4 probes
Aug 22, 2002 Sigcomm 2002
Expected Search Size (ESS)
• Allocation : p1, p2, p3,…, pm i pi = 1 ith item is allocated pi fraction of
storage. (keys placed in pi fraction of hosts)
• m items with relative query rates
q1 > q2 > q3 > … > qm. i qi = 1
• Search size for ith item is a Geometric r.v. with mean Ai = 1/( pi ).
• ESS is i qi Ai = (i qi / pi)/
Aug 22, 2002 Sigcomm 2002
Uniform and Proportional Replication
Two natural strategies:• Uniform Allocation: pi = 1/m
•Simple, resources are divided equally• Proportional Allocation: pi = qi
•“Fair”, resources per item proportional to demand• Reflects current P2P practices
Example: 3 items, q1=1/2, q2=1/3, q3=1/6
Uniform Proportional
Aug 22, 2002 Sigcomm 2002
Basic Questions
• How do Uniform and Proportional allocations perform/compare ?
• Which strategy minimizes the Expected Search Size (ESS) ?
• Is there a simple protocol that achieves optimal replication in decentralized unstructured networks ?
Aug 22, 2002 Sigcomm 2002
Insoluble queries• Search always extends to the maximum
allowed search size.• If we fix the available storage for copies, the
query rate distribution, and the number if items that we wish to be “locatable”, then
• The maximum required search size depends on the smallest allocation of an item. Thus,
• Uniform allocation minimizes this maximum and thus the cost induced by insoluble queries. What about the cost of soluble queries? Answer is more surprising …
Aug 22, 2002 Sigcomm 2002
ESS under Uniform and Proportional Allocations
(soluble queries)• Lemma: The ESS under either Uniform or
Proportional allocations is m/– Independent of query rates (!!!)– Same ESS for Proportional and Uniform (!!!)
Proportional:ASS is (i qi / pi)/(i qi / qi)/m/
Uniform:ASS is (i qi / pi)/(i m qi)/m/i qi m/
• Proof…
Aug 22, 2002 Sigcomm 2002
Space of Possible AllocationsDefinition: Allocation p1, p2, p3,…, pm is “in-between”
Uniform and Proportional if for 1i <m, q i+1/q i < p i+1/p i < 1
Theorem1: All (strictly) in-between strategies are (strictly) better than Uniform and Proportional
Theorem2: p is worse than Uniform/Proportional if for all i, p i+1/p i > 1 (more popular gets less) OR for all i, q i+1/q i > p i+1/p i (less popular gets less than
“fair share”)
Proportional and Uniform are the worst “reasonable” strategies (!!!)
Aug 22, 2002 Sigcomm 2002
q2/q1
p2/p
1Space of allocations on 2 items
Worse than prop/uniMore popular item gets less.
Worse than prop/uni
More popular gets more thanits proportional share
Better than prop/uni
Uniform
Proportional
SR
Aug 22, 2002 Sigcomm 2002
So, what is the best strategy for soluble queries ?
Aug 22, 2002 Sigcomm 2002
Square-Root Allocationpi is proportional to square-root(qi)
m
jj
ii
q
qp
1
• Lies “In-between” Uniform and Proportional
• Theorem: Square-Root allocation minimizes the ESS (on soluble queries)
Minimize i qi / pi such that i pi = 1
Aug 22, 2002 Sigcomm 2002
How much can we gain by using SR ?wi iq Zipf-like query rates
Aug 22, 2002 Sigcomm 2002
OK• SR is best for soluble queries• Uniform minimizes cost of insoluble queries
OPT is a hybrid of Uniform and SR
Tuned to balance cost of soluble and insoluble queries.
What is the optimal strategy?
Aug 22, 2002 Sigcomm 2002
UniformSR
10^4 items, Zipf-like w=1.5
All Soluble
85% Soluble
All Insoluble
Aug 22, 2002 Sigcomm 2002
We now know what we need.
How do we get there?
Aug 22, 2002 Sigcomm 2002
Replication Algorithms
• Fully distributed where peers communicate through random probes; minimal bookkeeping; and no more communication than what is needed for search.
• Converge to/obtain SR allocation when query rates remain steady.
• Uniform and Proportional are “easy” :-– Uniform: When item is created, replicate its key
in a fixed number of hosts.– Proportional: for each query, replicate the key
in a fixed number of hosts
Desired properties of algorithm:
Aug 22, 2002 Sigcomm 2002
Model for Copy Creation/Deletion
• Creation: after a successful search, C(s) new copies are created at random hosts.
• Deletion: is independent of the identity of the item; copy survival chances are non-decreasing with creation time. (i.e., FIFO at each node)
<Ci> average value of C used to replicate ith item.
Claim: If <Ci>/<Cj> remains fixed over time, and <Ci>, <Cj> , then pi/pj qi <Ci>/qj <Cj>
Property of the process:
Aug 22, 2002 Sigcomm 2002
Creation/Deletion Process
ii qC 1 jiji qqpp If then
Corollary:
Algorithm for square-root allocation needs to have <Ci> equal to or converge to a value inversely proportional to
iq
Aug 22, 2002 Sigcomm 2002
SR Replication Algorithms• Path replication: number of new copies C(s) is
proportional to the size of the search (Freenet)– Converges to SR allocation (+reasonable conditions)– Convergence unstable with delayed creations
• Sibling memory: each copy remembers the number of sibling copies,– Quickly “on target”– For “good estimates” need to find several copies.
• Probe memory: each peer records number and combined search size of probes it sees for each item. C(S) is determined by collecting this info from number of peers proportional to search size. – Immediately “on target”– Extra communication (proportional to that needed for search).
Aug 22, 2002 Sigcomm 2002
Alg1: Path Replication• Number of new copies produced per query,
<Ci>, is proportional to search size 1/pi
• Creation rate is proportional to qi <Ci>• Steady state: creation rate proportional to
allocation pi, thus
iiiii ppqCq
ii qp
Aug 22, 2002 Sigcomm 2002
Simulation
Path replicationSibling number
Host
s w
ith c
op
y
time
Delay = 0.25 * copy lifetime; 10000 hosts
Aug 22, 2002 Sigcomm 2002
Summary• Random Search/replication Model: probes to
“random” hosts• Proportional allocation – current practice• Uniform allocation – best for insoluble queries• Soluble queries:
• Proportional and Uniform allocations are two extremes with same average performance
• Square-Root allocation minimizes Average Search Size
• OPT (all queries) lies between SR and Uniform• SR/OPT allocation can be realized by simple
algorithms.
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