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Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Motivations and Objectives Capabilities are unequal in P2P networks Peers are heterogeneous Distributions of bandwidth, processing power, uptime, storage space… follow a heavy tailed curve Issue: Allocating Resources in a clever way GOAL: Classifying nodes into categories, slices Based on individual characteristics: attributes Typically, answering the question:

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Motivations and Objectives Capabilities are unequal in P2P networks Peers are heterogeneous Distributions of bandwidth, processing power, uptime, storage space… follow a heavy tailed curve Issue: Allocating Resources in a cleaver way GOAL: Classifying nodes into categories Based on individual characteristics: attributes Typically, answering the question: HOW CAPABLE, POWERFUL… AM I w.r.t. OTHERS?

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Classifying the system nodes

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Classifying the system nodes …using their attribute values (assume a single attribute for simplicity reason)

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Classifying the system nodes Attribute values

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Classifying the system nodes Attribute values Normalized Indices pi

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal #4#3#2#1 Classifying the system nodes Attribute Values ai Normalized Indices pi 0 1 Slices

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Model System is large and dynamic: Contains n nodes: n of the order of a million. Nodes join and leave the system at any time. Nodes may crash too. Each node i has an attribute value ai, knows the slices (ex: 10 equally sized slices of size, each containing 10% of the nodes), and maintains a communication view Vi : A constant (or log. in n ) number of neighbors j, Their position estimate pj’, their attribute aj, (and their age), Communication occurs only through neighbors.

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Previous Result: Jelasity and Kermarrec 2006 The communication overlay is dynamic Neighborhood is provided by the Newscast protocol. Each node i randomly selects a position estimate pi’ (estimate of the normalized index) pi’ is a real drawn uniformly into (0;1] Each node i periodically: Exchanges its estimate with one neighbor j if i and j are misplaced: i.e., (ai - aj)(pi’ – pj’) < 0. Result: the system gets ordered exponentially fast (in the number of exchanges) For any node couple i and j, ai pi’

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal What means efficiency? Global Disorder Measure [JK06]: GDM = ∑ j (indexOf(pj’) – indexOf(pj))² Sum of squared difference between the estimated position and the right position. Slice Disorder Measure: SDM = ∑ j |sj’ – sj| Sum of distances between the estimated slice and the right slice. sj’ = 1 ? sj = pj’ = 4/11 ? pj = 5/11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal What means efficiency? Same experiment evaluated using the: 1)GDM: global disorder measure and 2)SDM: slice disorder measure.

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal What means efficiency? Same experiment evaluated using the: 1)GDM: global disorder measure and 2)SDM: slice disorder measure. SDM gets stuck!

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering Similar to [JK06] Uses Local Disorder Measure LDM LDM(i) = ∑ j in Vi (indexOf(pj’) – indexOf(pj))² Protocol Loop { 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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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering Similar to [JK06] Uses Local Disorder Measure LDM LDM(i) = ∑ j in Vi (pj’ - pj)² Protocol Loop { 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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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering /11 9/11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering /11 7/11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering /11 2/11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering /11 4/11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 1: Ordering

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Result: Slight convergence speed up n = 10 4 #slices = 100 |V| = 20

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal 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 1 Slices Normalized indices

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Problem: wrong slice assignment k = #slices, the size of any slice is Δ in (0,1]. X i, rand. var., = #peers that estimate their slice as S i E[X i ] = Δn. Probability that slices have expected size ( ±β ). Pr[ ∀ j, |X j – Δn| ≤ βΔn] ≤ 1 - 2ke -O( β 2 Δn). Provided this, the worst case scenario is: O(βΔk 2 n) nodes can not identify their slice. Example ( n=10 6, k=10, β=0.01 ): about 10% of the system can not identify their slice.

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking No random values! Protocol Loop { Update/Shuffle view Vi using an underlying protocol. l += #neighbors with lower attribute value. g += #neighbors Sends ai to a randomly chosen neighbor Sends ai to the neighbor that is the closest to a slice boundary Update slice assignment si’ = s : l/g in s. } Upon reception { Receive aj from j if ( aj < ai ) l += 1; g +=1 ; Update slice assignment si’ = s : l/g in s. }

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking No random values! Protocol Loop { Update/Shuffle view Vi using an underlying protocol. l += #neighbors with lower attribute value. g += #neighbors Sends ai to a randomly chosen neighbor Sends ai to the neighbor that is the closest to a slice boundary Update slice assignment si’ = s : l/g in s. } Upon reception { Receive aj from j if ( aj < ai ) l += 1; g +=1 ; Update slice assignment si’ = s : l/g in s. } Same number of messages

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking /11

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking /2

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking /4

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Algorithm 2: Ranking /3

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Result 1: Unlimited convergence n = 10 4 #slices = 100 |V| = 20

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Result 1: Unlimited convergence Ranking precision keeps improving n = 10 4 #slices = 100 |V| = 20

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Result 2: Feasibility

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Result 3: 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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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Performance analysis d, is the distance from p i ’ to the closest slice boundary. For confidence coefficient of 99,99%, the required number of attribute value drawn is m i ≥ z p i ’ (1 – p i ’) / d 2, with z <16, a constant.

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal Conclusion Churn-tolerant algorithm Gossip-based mechanisms. Slice belongingness re-approximation. Scalable algorithm Limited number of neighbors. Size of the system is unknown. Applications Δ –approx. of the 1-dimensional k -centroid problem. (A step towards the conjecture of K.Birman at Leiden’06) => “Facility location”. “Resource allocation”, “Super-peers identification”… Future work d -dimensional extension (with many attributes). Gossip-based quorums: one node of each slice, vs. all nodes of 1 slice.

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TR 1829 IRISA January, 4th Fernandez, Gramoli, Jimenez, Kermarrec, Raynal References Ordered Slicing of Very Large-Scale Overlay Networks M. Jelasity and A.-M. Kermarrec In Proc. of the 6 th IEEE Conference on P2P Computing, Randomized Algorithms R. Motwani, P. Raghavan Cambridge University Press, 1995 Time Bounds for Selection M. Blum, R. Floyd, V. Pratt, R. Rivest, and R. Tarjan Journal Computer and System Sciences 7: , 1972

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