Distributed Slicing in Dynamic Systems

Distributed Slicing in Dynamic Systems

A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal

ICDCS 2007June

Fernandez, Gramoli, Jimenez, Kermarrec, Raynal

Context

Distributed System

Interconnected set of nodes

Heterogeneous environment

Resources are heterogeneously spread• Bandwidth

• Processing Power

• Storage Space

• Uptime

• …

Large-scale dynamic environment

Nodes leave and join at any time

Large amount of nodes

No global information

1,50,3

1

10098

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Problem

What means rich/poor in this context?

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Problem


20

Am I rich or poor?

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Problem


202

6

15

rich?

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Problem


20300

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815220

poor?

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Applications

Remedy the Gnutella problem in Peer-to-peer Gnutella performance was limited by poor nodes

Kazaa/Skype sollicitate rich nodes rather than poor nodes

Allocating resources/nodes to services Streaming service needs nodes with highest bandwidth Non-critical service can run on unstable nodes File-sharing service requires nodes with many files …

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Objectives

Classifying nodes into categories, slices Based on individual characteristics: attributes A slice corresponds to a portion of the system

Typically, answering the question:

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Objectives

HOW RICH AM I COMPARED TO OTHERS?

Classifying nodes into categories, slices Based on individual characteristics: attributes A slice corresponds to a portion of the system

Typically, answering the question:

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Classifying the system nodes

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…using their attribute values (assume a single attribute for simplicity reason)

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0 100

Attribute values ai

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0 100

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Attribute values ai

NormalizedIndices pi

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#4#3#2#1


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0 100

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NormalizedIndices pi

0 1Slices si

Attribute values ai

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ModelDynamic system System of n nodes Nodes join and leave the system at any time Nodes may crash too

Each node i knows its attribute value ai, its position estimate pi’, the slices (ex: 10 equally sized slices, each containing

10% of the nodes), a communication view Vi:

• constant number of neighbors j,• their position estimate pj’, their attribute aj, (and their age)

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Underlying Gossip-based Overlay

Each node i periodically: Exchanges information with its neighbors (nodes of its

view)• Views, values…

Computes its new view and its new state• View, value…

Dynamic overlay: Failed nodes are naturally removed from views Joining nodes are naturally added into views

Scalable overlay: Limited amount of information stored Limited amount of information exchanged

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Jelasity, Kermarrec 2006 (JK)

Each node i sets pi’ as a uniform random value

Each node i looks for a misplaced neighbor j A neighbor j is misplaced with i iff (ai – aj)(pi’ – pj’) < 0

Then, nodes i and j exchange pi’ and pj’

The sequence of position estimates matches the sequence of attribute values

For any node couple i and j, ai<aj <=> pi’<pj’

Convergence to this ordering is exponentially fast

(in the number of execution)

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Algorithm 1: Ordering

Similar to JK

Uses Local Disorder Measure LDM Let lpj’ (resp. lpj) be the normalized index of j random value

(resp. attribute value) among all j of Vi i

LDM(i) = ∑ j in Vi i (lpj’ – lpj)²

ProtocolDo periodically {

Update view Vi using an underlying protocol.Choose the neighbor j that minimizes the LDM(i).Exchange random values pi’ and pj’ Update the random value pi’ w/ pj’ if necessary.Update slice assignment si’ := s : pi’ in s.

}

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Similar to JK

Uses Local Disorder Measure LDM Let lpj’ (resp. lpj) be the normalized index of j random value (resp.

attribute value) among all j of Vi i

LDM(i) = ∑ j in Vi i (lpj’ – lpj)²


Update view Vi using an underlying protocol.Choose the neighbor j that minimizes the LDM(i).Exchange random values pi’ and pj’. Update the random value pi’ w/ pj’ if necessary.Update slice assignment si’ := s : pi’ in s.

}

Difference with JK

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4/11 9/11

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Result: Slight convergence speed up

n = 104

#slices = 10|V| = 20

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If random values are not perfectly uniformly distributed

…some nodes might never find their slice e.g. the 3 nodes of S2 in the example above

Problem: Wrong Slice Assignment

#4#3#2#1

0 1Slices

Normalizedindices

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Algorithm 2: RankingNo random values!


Update/Shuffle view Vi using an underlying protocol.l += #neighbors with lower attribute value.g += #neighborsSends ai to a randomly chosen neighborSends ai to the neighbor that is the closest to a slice boundary

Update slice assignment si’ = s such that l/g in s.}Upon reception {

Receive aj from jif (aj < ai) l += 1;

g+=1 ;Update slice assignment si’ = s such that l/g in s.

}

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Algorithm 2: RankingNo random values!


Update/Shuffle view Vi using an underlying protocol.l += #neighbors with lower attribute value.g += #neighborsSends ai to a randomly chosen neighborSends ai to the neighbor that is the closest to a slice boundary

Update slice assignment si’ = s such that l/g in s.}Upon reception {

Receive aj from jif (aj < ai) l += 1; g+=1 ;Update slice assignment si’ = s such that l/g in s.

}

Same number of messages

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Algorithm 2: Ranking

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0/2

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Result 1: Unlimited convergence

n = 104

#slices = 100|V| = 20

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Result 1: Unlimited convergence

Ranking precision keeps improving

n = 104

#slices = 100|V| = 20

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Result 2: Tolerating Dynamism

Churn is correlated with attribute values!

e.g. the attribute is the remaining batery lifetime or available storage space.

Churn

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Performance Analysis

d, is the distance from pi’ to the closest slice boundary.For confidence coefficient of 99,99%, the required number of attribute value drawn is

mi ≥ z pi’ (1 – pi’) / d2,

with z <16, a constant.

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Conclusion

Churn-tolerant algorithmGossip-based mechanisms.Slice belongingness re-approximation.

Scalable algorithmLimited number of neighbors.Size of the system is unknown.

Applications of the distributed slicingResource allocationSupernodes / Ultrapeers election

Future workCan we obtain convergence speed of the first algorithm with the

accuracy of the second algorithm?

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References

Ordered Slicing of Very Large-Scale Overlay NetworksM. Jelasity and A.-M. Kermarrec In Proc. of the 6th IEEE Conference on P2P Computing, 2006.

Randomized AlgorithmsR. Motwani, P. RaghavanCambridge University Press, 1995

Time Bounds for SelectionM. Blum, R. Floyd, V. Pratt, R. Rivest, and R. TarjanJournal Computer and System Sciences 7:448-461, 1972

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Result 3: Feasibility

Distributed Slicing in Dynamic Systems

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