1 Distributed k -ary System Algorithms for Distributed Hash Tables Ali Ghodsi aligh@kth.se ali/thesis/ PhD Defense, 7th December 2006,

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Distributed k-ary SystemAlgorithms for Distributed Hash Tables

Ali Ghodsialigh@kth.se

http://www.sics.se/~ali/thesis/

PhD Defense, 7th December 2006, KTH/Royal Institute of Technology

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Distributed k-ary SystemAlgorithms for Distributed Hash Tables

Ali Ghodsialigh@kth.se

http://www.sics.se/~ali/thesis/

PhD Defense, 7th December 2006, KTH/Royal Institute of Technology

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Presentation Overview

• Gentle introduction to DHTs

• Contributions

• The future

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What’s a Distributed Hash Table (DHT)?

• An ordinary hash table

• Every node provides a lookup operation•Provide the value associated with a key

• Nodes keep routing pointers•If item not found, route to another node

Key ValueAlexander

Berlin

Ali Stockholm

Marina Gothenburg

PeterLouvain la neuve

Seif Stockholm

Stefan Stockholm

, which is distributed

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So what?

•Characteristic properties•Scalability

•Number of nodes can be huge•Number of items can be huge

•Self-manage in presence joins/leaves/failures•Routing information •Data items

Time to find data is logarithmic

Size of routing tables is logarithmic

Example:

log2(1000000)≈20

EFFICIENT!

Store number of items proportional to number of nodes

Typically:

With D items and n nodes

Store D/n items per node

Move D/n items when nodes join/leave/fail

EFFICIENT!

Self-management routing info:• Ensure routing information

is up-to-date

Self-management of items:• Ensure that data is always

replicated and available

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Presentation Overview

• …• …

• What’s been the general motivation for DHTs?

• …• …

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Traditional Motivation (1/2)

• Peer-to-Peer filesharing very popular

• Napster• Completely centralized• Central server knows who has what• Judicial problems

• Gnutella• Completely decentralized• Ask everyone you know to find

data• Very inefficient

central index

decentralized index

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Traditional Motivation (2/2)• Grand vision of DHTs

• Provide efficient file sharing

• Quote from Chord: ”In particular, [Chord] can help avoid single points of failure or control that systems like Napster possess, and the lack of scalability that systems like Gnutella display because of their widespread use of broadcasts.” [Stoica et al. 2001]

• Hidden assumptions• Millions of unreliable nodes• User can switch off computer any time (leave=failure)• Extreme dynamism (nodes joining/leaving/failing)• Heterogeneity of computers and latencies• Unstrusted nodes

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Our philosophy

• DHT is a useful data structure

• Assumptions might not be true• Moderate amount of dynamism• Leave not same thing as failure

• Dedicated servers• Nodes can be trusted• Less heterogeneity

• Our goal is to achieve more given stronger assumptions

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Presentation Overview

• …• …

• How to construct a DHT?

• …• …

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How to construct a DHT (Chord)?

• Use a logical name space, called the identifier space, consisting of identifiers {0,1,2,…, N-1}

• Identifier space is a logical ring modulo N

• Every node picks a random identifier

• Example:

• Space N=16 {0,…,15}

• Five nodes a, b, c, d• a picks 6• b picks 5• c picks 0• d picks 5• e picks 2

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Definition of Successor

• The successor of an identifier is the first node met going in clockwise direction starting at the identifier

• Example• succ(12)=14• succ(15)=2• succ(6)=6

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Where to store data (Chord) ?

•Use globally known hash function, H

•Each item <key,value> gets identifier H(key)

•Store each item at its successor•Node n is responsible for item k

•Example• H(“Marina”)=12• H(“Peter”)=2• H(“Seif”)=9• H(“Stefan”)=14

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Key Value

Alexander Berlin

MarinaGothenburg

PeterLouvain la neuve

Seif Stockholm

Stefan Stockholm

Store number of items proportional to number of nodes

Typically:

With D items and n nodes

Store D/n items per node

Move D/n items when nodes join/leave/fail

EFFICIENT!

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Where to point (Chord) ?•Each node points to its successor

•The successor of a node n is succ(n+1)•Known as a node’s succ pointer

•Each node points to its predecessor•First node met in anti-clockwise direction starting at n-1 •Known as a node’s pred pointer

•Example• 0’s successor is succ(1)=2• 2’s successor is succ(3)=5• 5’s successor is succ(6)=6• 6’s successor is succ(7)=11• 11’s successor is succ(12)=0

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DHT Lookup

•To lookup a key k

• Calculate H(k)

• Follow succ pointers until item k is found

•Example• Lookup ”Seif” at node 2

• H(”Seif”)=9

• Traverse nodes:• 2, 5, 6, 11 (BINGO)

• Return “Stockholm” to initiator

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Key Value

Alexander Berlin

Marina Gothenburg

PeterLouvain la neuve

Seif Stockholm

Stefan Stockholm

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Speeding up lookups

• If only pointer to succ(n+1) is used• Worst case lookup time is N, for N nodes

• Improving lookup time• Point to succ(n+1)• Point to succ(n+2)• Point to succ(n+4)• Point to succ(n+8)• …• Point to succ(n+2M)

• Distance always halved to the destination

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Time to find data is logarithmic

Size of routing tables is logarithmic

Example:

log2(1000000)≈20

EFFICIENT!

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Dealing with failures

• Each node keeps a successor-list• Pointer to f closest successors

• succ(n+1)• succ(succ(n+1)+1)• succ(succ(succ(n+1)+1)+1)• ...

• If successor fails• Replace with closest alive

successor

• If predecessor fails• Set pred to nil

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Handling Dynamism

• Periodic stabilization used to make pointers eventually correct

• Try pointing succ to closest alive successor

• Try pointing pred to closest alive predecessor

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Presentation Overview

• Gentle introduction to DHTs

• Contributions

• The future

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Outline

• …• …

• Lookup consistency

• …• …

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Problems with periodic stabilization

• Joins and leaves can result in inconsistent lookup results

• At node 12, lookup(14)=14

• At node 10, lookup(14)=15

1012 14 15

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Problems with periodic stabilization

• Leaves can result in routing failures

1013

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Problems with periodic stabilization

• Too many leaves destroy the system

• #leaves+#failures/round < |successor-list|

10 11 12 14 15

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Outline

• …• …

• Atomic Ring Maintenance

• …• …

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Atomic Ring Maintenance

• Differentiate leaves from failures

• Leave is a synchronized departure

• Failure is a crash-stop

• Initially assume no failures• Build a ring initially

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Atomic Ring Maintenance

• Separate parts of the problem• Concurrency control

• Serialize neighboring joins/leaves

• Lookup consistency

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Naïve Approach

• Each node i hosts a lock called Li

• For p to join or leave:• First acquire Lp.pred

• Second acquire Lp

• Third acquire Lp.succ

• Thereafter update relevant pointers

• Can lead to deadlocks

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Our Approach to Concurrency Control

• Each node i hosts a lock called Li

• For p to join or leave:• First acquire Lp

• Thereafter acquire Lp.succ

• Thereafter update relevant pointers

• Each lock has a lock queue• Nodes waiting to acquire the lock

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Safety

• Non-interference theorem:

•When node p acquires both locks:

•Node p’s successor cannot leave

•Node p’s ”predecessor” cannot leave

•Other joins cannot affect ”relevant” pointers

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Dining Philosophers

• Problem similar to the Dining philosophers’ problem

• Five philosophers around a table• One fork between each philosopher

(5)• Philosophers eat and think• To eat:

• grab left fork• then grab right fork

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Deadlocks

• Can result in a deadlock• If all nodes acquire their first lock• Every node waiting indefinitely for second lock

• Solution from Dining philosophers’• Introduce asymmetry• One node acquires locks in reverse order

• Node with highest identifier reverses• If n<n.succ, then n has highest identity

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Pitfalls• Join adds node/“philosopher”

• Solution: some requests in the lock queue forwarded to new node

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12 12

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Pitfalls

• Leave removes a node/“philosopher”• Problem:

if leaving node gives lock queue to its successor, nodes can get worse position in queue: starvation

• Use forwarding to avoid starvation• Lock queue empty after local leave

request

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Correctness

• Liveness Theorem: • Algorithm is starvation free

• Also free from deadlocks and livelocks

• Every joining/leaving node will eventually succeed getting both locks

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Performance drawbacks

• If many neighboring nodes leaving• All grab local lock• Sequential progress

• Solution• Randomized locking• Release locks and retry• Liveness with high probability

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Lookup consistency: leaves

• So far dealt with concurrent joins/leaves• Look at concurrent join/leaves/lookups

• Lookup consistency (informally):• At any time, only one node responsible

for any key

• Joins/leaves should “not affect” functionality of lookups

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Lookup consistency

• Goal is to make joins and leaves appear as if they happened instantaneously

• Every leave has a leave point• A point in global time, where the whole

system behaves as if the node instantaneously left

• Implemented with a LeaveForward flag• The leaving node forwards messages to

successor if LeaveForward is true

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Leave Algorithm

pred:=p

succ:=r

LeaveForward=true

LeaveForward=false

<UpdateSucc, succ=r>

Node p Node q (leaving) Node r

<StopForwarding>

<LeavePoint, pred=p>

leave point

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Lookup consistency: joins

• Every join has a join point• A point in global time, where the whole

system behaves as if the node instantaneously joined

• Implemented with a JoinForward flag• The successor of a joining node

forwards messages to new node if JoinForward is true

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Join Algorithm

Join Point

JoinForward=trueoldpred=predpred=q

JoinForwarding=false

succ:=q

pred:=psucc:=r

<JoinPoint, pred=p>

<UpdateSucc,

succ=q>

Node p Node q (joining) Node r

<StopForwarding>

<Finish>

<UpdatePred, pred=q>

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Outline

• …• …

• What about failures?

• …• …

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Dealing with Failures

• We prove it is impossible to provide lookup consistency on the Internet

• Assumptions• Availability (always eventually answer)• Lookup consistency• Partition tolerance

• Failure detectors can behave as if the networked partitioned

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Dealing with Failures

• We provide fault-tolerant atomic ring• Locks leased• Guarantees locks are always released

• Periodic stabilization ensures • Eventually correct ring• Eventual lookup consistency

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Contributions

• Lookup consistency in presence of joins/leaves• System not affected by joins/leaves• Inserts do not “disappear”

• No routing failures when nodes leave

• Number of leaves not bounded

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Related Work

• Li, Misra, Plaxton (’04, ’06) have a similar solution

• Advantages• Assertional reasoning• Almost machine verifiable proofs

• Disadvantages• Starvation possible• Not used for lookup consistency• Failure-free environment assumed

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Related Work

• Lynch, Malkhi, Ratajczak (’02), position paper with pseudo code in appendix

• Advantages• First to propose atomic lookup consistency

• Disadvantages• No proofs• Message might be sent to a node that left• Does not work for both joins and leaves

together• Failures not dealt with

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Outline

• …• …

• Additional Pointers on the Ring

• …• …

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Routing

• Generalization of Chord to provide arbitrary arity

• Provide logk(n) hops per lookup

•k being a configurable parameter

•n being the number of nodes

• Instead of only log2(n)

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Achieving logk(n) lookup

Interval 1

Interval 2

Interval 3

Interval 0

0

32

48

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8

12

16

I3I2I1I0Node 0

48…63

32…47

16…31

0…15Level 1

• Each node logk(N) levels, N=kL

• Each level contains k intervals,

• Example, k=4, N=64 (43), node 0

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I3I2I1I0Node 0

48…63

32…47

16…31

0…15Level 1Interval 2

Interval 1

Interval 3

Interval 0

Achieving logk(n) lookup

0

32

48

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8

12

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• Each node logk(N) levels, N=kL

• Each level contains k intervals,

• Example, k=4, N=64 (43), node 0

I3I2I1I0Node 0

12…15

8…114…70…3Level 2

48…63

32…47

16…31

0…15Level 1

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I3I2I1I0Node 0

12…15

8…114…70…3Level 2

48…63

32…47

16…31

0…15Level 1

Achieving logk(n) lookup

0

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48

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I3I2I1I0Node 0

3210Level 3

12…15

8…114…70…3Level 2

48…63

32…47

16…31

0…15Level 1

• Each node logk(N) levels, N=kL

• Each level contains k intervals,

• Example, k=4, N=64 (43), node 0

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Arity important

• Maximum number of hops can be configured

• Example, a 2-hop system

rNNNr

r

NNk

rr

1

11 log)(log)(log

2)(log

N

Nk

N

rNk1

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•Each node has (k-1)logk(N) pointers• Node p’s pointers point at

Placing pointers

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)))1(mod)1((1()( ki

kkipif

0

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48

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Node 0’s pointersf(1)=1f(2)=2f(3)=3f(4)=4f(5)=8f(6)=12f(7)=16f(8)=32f(9)=48

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Greedy Routing

• lookup(i) algorithm

• Use pointer closest to i, without “overshooting” i

• If no such pointer exists, succ is responsible for i

i

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Routing with Atomic Ring Maintenance

• Invariant of lookup• Last hop is always predecessor of

responsible node

• Last step in lookup• If JoinForward is true, forward to pred• If LeaveForward is true, forward to

succ

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Avoiding Routing Failures

• If nodes leave, routing failures can occur

• Accounting algorithm • Simple Algorithm

• No routing failures of ordinary messages

• Fault-free Algorithm• No routing failures

• Many cases and interleavings• Concurrent joins and leaves,

pointers in both directions

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General Routing

• Three lookup styles

• Recursive

• Iterative

• Transitive

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Reliable Routing• Reliable lookup for each style

• If initiator doesn’t crash, responsible node reached• No redundant delivery of messages

• General strategy• Repeat operation until success• Filter duplicates using unique identifiers

• Iterative lookup• Reliability easy to achieve

• Recursive lookup• Several algorithms possible

• Transitive lookup• Efficient reliability hard to achieve

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Outline

• …• …

• One-to-many Communication

• …• …

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Group Communication on an Overlay

• Use existing routing pointers • Group communication

• DHT only provides key lookup• Complex queries by searching the overlay• Limited horizon broadcast• Iterative deepening

• More efficient than Gnutella-like systems• No unintended graph partitioning• Cheaper topology maintenance [castro04]

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Group Communication on an Overlay

• DHT builds a graph• Why not use general graph algorithms?

• Can use the specific structure of DHTs• More efficient• Avoids redundant messages

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Broadcast Algorithms

• Correctness conditions:• Termination

• Algorithm should eventually terminate

• Coverage• All nodes should receive the broadcast

message

• Non-redundancy• Each node receives the message at most once

• Initially assume no failures

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Naïve Broadcast

• Naive Broadcast Algorithmsend message to succ until:

initiator reached or overshooted

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initiator

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Naïve Broadcast

• Naive Broadcast Algorithmsend message to succ until:

initiator reached or overshooted

• Improvement• Initiator delegates half

the space to neighbor

• Idea applied recursively• log(n) time and n messages

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Simple Broadcast in the Overlay

• Dissertation assumes general DHT model

event n.SimpleBcast(m, limit) % initially limit = nfor i:=M downto 1 do

if u(i) ∈ (n,limit) thensendto u(i) : SimpleBcast(m, limit)limit := u(i)

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”Advanced” Broadcast• Old algorithm on k-ary trees

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Getting responses

• Getting a reply• Nodes send directly back to initiator• Not scalable

• Simple Broadcast with Feedback• Collect responses back to initiator• Broadcast induces a tree, feedback in reverse direction

• Similar to simple broadcast algorithm• Keeps track of parent (par)• Keeps track of children (Ack)• Accumulate feedback from children, send to parent

• Atomic ring maintenance• Acquire local lock to ensure nodes do not leave

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Outline

• …• …

• Advanced One-to-many Communication

• …• …

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Motivation for Bulk Operation

• Building MyriadStore in 2005• Distributed backup using the DKS DHT

• Restoring a 4mb file• Each block (4kb) indexed in DHT• Requires 1000 items in DHT

• Expensive• One node making 1000 lookups• Marshaling/unmarshaling 1000 requests

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Bulk Operation

• Define a bulk set: I• A set of identifiers

• bulk_operation(m, I)• Send message m to every node i ∈ I

• Similar correctness to broadcast• Coverage: all nodes with identifier in I • Termination• Non-redundancy

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Bulk Owner Operation with Feedback

• Define a bulk set: I• A set of identifiers

• bulk_own(m, I)• Send m to every node responsible for an

identifier i ∈ I

• Example• Bulk set I={4}• Node 4 might not exist• Some node is responsible for identifier 4

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Bulk Operation with Feedback

• Define a bulk set: I• A set of identifiers

• bulk_feed(m, I)• Send message m to every node i ∈ I• Accumulate responses back to initiator

• bulk_own_feed(m, I)• Send message m to every node

responsible for i ∈ I• Accumulate responses back to initiator

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Bulk Properties (1/2)

• No redundant messages

• Maximum log(n) messages per node

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Bulk Properties (2/2)• Two extreme cases

• Case 1• Bulk set is all identifiers• Identical to simple broadcast• Message complexity is n• Time complexity is log(n)

• Case 2• Bulk set is a singleton with one identifier• Identical to ordinary lookup• Message complexity is log(n)• Time complexity is in log(n)

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Pseudo Reliable Broadcast

• Pseudo-reliable broadcast to deal with crash failures

• Coverage property• If initiator is correct, every node gets the message

• Similar to broadcast with feedback

• Use failure detectors on children• If child with responsibility to cover I fails• Use bulk to retry covering interval I

• Filter redundant messages using unique identifiers

• Eventually perfect failure detector for termination• Inaccuracy results in redundant messages

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Applications of bulk operation

• Bulk operation• Topology maintenance: update nodes in bulk

set• Pseudo-reliable broadcast: re-covering intervals

• Bulk owner• Multiple inserts into a DHT

• Bulk owner with feedback• Multiple lookups in a DHT• Range queries

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Outline

• …• …

• Replication

• …• …

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Successor-list replication

• Successor-list replication• Replicate a node’s item on its f

successors• DKS, Chord, Pastry, Koorde etcetera.

• Was abandoned in favor of symmetric replication because …

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Motivation: successor-lists• If a node joins or leaves

• f replicas need to be updated

Color represents data item

Replication degree 3

Every color replicated three times

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Motivation: successor-lists• If a node joins or leaves

• f replicas need to be updated

Color represents data item

Node leaves

Yellow, green, red, blue need to be re-distributed

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Multiple hashing• Rehashing

• Store each item <k,v> at• succ( H(k) )• succ( H(H(k)) )• succ( H(H(H(k))) )• …

• Multiple hash functions• Store each item <k,v> at

• succ( H1(k) )• succ( H2(k) )• succ( H3(k) )• …

• Advocated by CAN and Tapestry

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Motivation: multiple hashing

• Example• Item <”Seif”, ”Stockholm”>

• H(”Seif”)=7• succ(7)=9

• Node 9 crashes• Node 12 should get item from replica• Need hash inverse H-1(7)=”Seif” (impossible)• Items dispersed all over nodes (inefficient)

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Seif, Stockholm

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Symmetric Replication•Basic Idea

•Replicate identifiers, not nodes

•Associate each identifier i with f other identifiers:

•

•Identifier space partitioned into m

equivalence classes •Cardinality of each class is f, m=N/f

•Each node replicates the equivalence class of all identifiers it is responsible for

fkfN

kikr 0for,)(

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Symmetric replicationReplication degree f=4, Space={0,…,15}• Congruence classes modulo 4:

• {0, 4, 8, 12}• {1, 5, 9, 13}• {2, 6, 10, 14}• {3, 7, 11, 15}

0 12

1514

13 3

12

11

4

5

6

9 87

10

Data: 15, 0

Data: 1, 2, 3

Data: 4, 5

Data: 14, 13, 12, 11

Data: 6, 7, 8, 9, 10

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Ordinary ChordReplication degree f=4, Space={0,…,15}• Congruence classes modulo 4

• {0, 4, 8, 12}• {1, 5, 9, 13}• {2, 6, 10, 14}• {3, 7, 11, 15}

0 12

1514

13 3

12

11

4

5

6

9 87

10

Data: 15, 0

Data: 1, 2, 3

Data: 4, 5

Data: 14, 13, 12, 11

Data: 10, 9, 8, 7

Data: 11, 12

Data: 13, 14, 15

Data: 0, 1

Data: 2, 3, 4, 5, 6

Data: 6, 5, 4, 3

Data: 2, 1, 0, 15Data: 7, 8

Data: 3, 4

Data: 9, 10, 11

Data: 5, 6, 7

Data: 12, 13

Data: 8, 9

Data: 14, 15, 0, 1, 2

Data: 10, 11, 12, 13, 14

Data: 6, 7, 8, 9, 10

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Data: 15, 0

Data: 1, 2, 3

Data: 4, 5

Data: 14, 13, 12, 11

Data: 6, 7, 8, 9, 10

Data: 11, 12

Cheap join/leaveReplication degree f=4, Space={0,…,15}• Congruence classes modulo 4

• {0, 4, 8, 12}• {1, 5, 9, 13}• {2, 6, 10, 14}• {3, 7, 11, 15}

0 12

1514

13 3

12

11

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5

6

9 87

10

Data: 10, 9, 8, 7

Data: 11, 12

Data: 13, 14, 15

Data: 0, 1

Data: 2, 3, 4, 5, 6

Data: 6, 5, 4, 3

Data: 2, 1, 0, 15Data: 7, 8

Data: 3, 4

Data: 9, 10, 11

Data: 5, 6, 7

Data: 12, 13

Data: 8, 9

Data: 14, 15, 0, 1, 2

Data: 10, 11, 12, 13, 14

Data: 7, 8

Data: 3, 4

Data: 0, 15

Data: 11, 12, 7, 8, 3, 4, 0, 15

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Contributions

• Message complexity for join/leave O(1)• Bit complexity remains unchanged

• Handling failures more complex• Bulk operation to fetch data• On average log(n) complexity

• Can do parallel lookups• Decreasing latencies• Increasing robustness• Distributed voting• Erasure codes

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Presentation Overview

• …• …

• Summary

• …• …

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Summary (1/3)

• Atomic ring maintenance• Lookup consistency for j/l• No routing failures as nodes j/l• No bound on number of leaves• Eventual consistency with failures

• Additional routing pointers• k-ary lookup• Reliable lookup• No routing failures with additional

pointers

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Summary (2/3)• Efficient Broadcast

• log(n) time and n message complexity• Used in overlay multicast

• Bulk operations• Efficient parallel lookups• Efficient range queries

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Summary (3/3)

• Symmetric Replication• Simple, O(1) message complexity for j/l

•O(log f) for failures

• Enables parallel lookups• Decreasing latencies• Increasing robustness• Distributed voting

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Presentation Overview

• Gentle introduction to DHTs

• Contributions

• The future

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Future Work (1/2)

• Periodic stabilization• Prove it is self-stabilizing

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Future Work (2/2)

• Replication Consistency• Atomic consistency impossible in

asynchronous systems• Assume partial synchrony• Weaker consistency models?• Using virtual synchrony

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Speculative long-term agenda

• Overlay today provides• Dynamic membership• Identities (max/min avail)• Only know subset of nodes• Shared memory registers

• Revisit distributed computing• Assuming an overlay as basic primitive• Leader election• Consensus• Shared memory consistency (started)• Transactions• Wave algorithms (started)

• Implement middleware providing these…

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Acknowledgments

• Seif Haridi• Luc Onana Alima

• Cosmin Arad• Per Brand• Sameh El-Ansary • Roland Yap

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THANK YOU

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Handling joins

• When n joins• Find n’s successor with lookup(n)• Set succ to n’s successor• Stabilization fixes the rest

Periodically at n:

1. set v:=succ.pred2. if v≠nil and v is in

(n,succ]3. set succ:=v4. send a notify(n) to succ

When receiving notify(p) at n:

1. if pred=nil or p is in (pred,n]

2. set pred:=p

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Handling leaves

• When n leaves• Just dissappear (like failure)

• When pred detected failed• Set pred to nil

• When succ detected failed• Set succ to closest alive in

successor list

Periodically at n:

1. set v:=succ.pred2. if v≠nil and v is in

(n,succ]3. set succ:=v4. send a notify(n) to succ

When receiving notify(p) at n:

1. if pred=nil or p is in (pred,n]

2. set pred:=p

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