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Lookup in Small Worlds -- A Survey --

Lookup in Small Worlds -- A Survey --. Pierre Fraigniaud CNRS, U. Paris Sud. Milgram’s Experiment. Source person s (e.g., in Wichita) Target person t (e.g., in Cambridge) Name, occupation, etc. Letter transmitted via a chain of individuals related on a personal basis

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Lookup in Small Worlds -- A Survey --

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  1. Lookup in Small Worlds -- A Survey -- Pierre Fraigniaud CNRS, U. Paris Sud

  2. Milgram’s Experiment • Source person s (e.g., in Wichita) • Target person t (e.g., in Cambridge) • Name, occupation, etc. • Letter transmitted via a chain of individuals related on a personal basis • Result: The “six degrees of separation”

  3. Augmented graphs Watts & Strogatz [Nature ‘98] H=(G,D) • Individuals as nodes of a graph G • Edges of G model relations between individuals deducible from their societal positions • D = probabilistic distribution • “Long links” = links added to G at random, according to D • Long links model relations between individuals that cannot be deduced from their societal positions

  4. v Augmented meshes Kleinberg [STOC ‘00] Meshes augmented with d-harmonic links u prob(uv) ≈ 1/dist(u,v)d Exactly 1 long link per node

  5. Greedy Routing • Source s = (s1,s2,…,sd) • Target t = (t1,t2,…,td) • Current node x selects among its 2d+1 neighbors the closest to t in the mesh (i.e., according to the Manhattan distance)

  6. long link long link Performances of Greedy Routing B=ball radius m/2 t distG(x,t)=m x  O(log n) expect. #steps to enter B  O(log2n) expect. #steps to reach t from s

  7. Performances of greedy routing Theorem(Kleinberg [STOC ’00]) Greedy routing performs in O(log2n) expected #steps in d-dimensional meshes augmented with d-harmonic distribution. Application: DHT “Symphony” (Manku, Bawa, Raghavan [USENIX ’03]) Can we improve this bound?

  8. Adding more long links Theorem(Kleinberg [STOC ’00]) In d-dimensional meshes augmented with c long links per node (chosen according to the d-harmonic distribution), greedy routing performs in O(log2n/c) expected #steps. In particular: c = log n O(log n) steps

  9. Bad news Theorem(Kleinberg [STOC ’00]) Greedy routing in d-dimensional meshes augmented with a k-harmonic distribution, k≠d, performs in Ω(nβ) expected #steps. Can we do better using the d-harmonic distribution?

  10. Yet another bad news Theorem(Barrière, F., Kranakis, Krizanc [DISC ’01]) Greedy routing in d-dimensional meshes augmented with the d-harmonic distribution performs in Ω(log2n) expected #steps. Can we do better using other distributions?

  11. Another bad news! Theorem(Aspnes, Diamadi, Shah [PODC’02]) Greedy routing in directed rings augmented with any distribution performs in Ω(log2n/loglog n) expected #steps. Probably true in undirected rings, and in higher dimensions… Is it the end of the game?

  12. A decentralized algorithm for routing Theorem(Lebhar, Schabanel [ICALP ’04]) There exists a distributed routing protocol that • Visits O(log2n) expected #nodes; • Discovers routes of expected length O(log n (loglog n)2).

  13. Applications DHT: • lookup in O(log2n) expected #steps • download in O(log n (loglog n)2) steps Does not apply to Milgram’s experiment (backtracks during the lookup)

  14. Increasing the awareness Neighbors-of-neighbors (NoN)

  15. Percolation theory 0 ≤ pi ≤ 1 withΣi pi = 1 • Kleinberg: for every node x, chose c edges (x,yi) with prob{(x,yi) is chosen} = pi • Remark: deg(x) = c • Percolation: for every edge (x,yi), prob{(x,yi) is in the network} = c pi prob{(x,yi) is not in the network} = 1 - c pi • Remark: E(deg(x)) = c

  16. Diameter of percolation graphsBenjamini, Berger [2000] Diameter D of rings: prob(x,y) = 1-e-β/dist(x,y)k ≈ β/dist(x,y)k With high probability: • k<1: D=O(1) • 1<k<2: D=O(logαn) α>0 • k>2: D=Ω(n)

  17. Diameter of percolation graphsCoppersmith, Gamarnik, Sviridenko [SODA ‘02] Diameter D of d-dimensional meshes: prob(x,y) = 1/dist(x,y)k With high probability: • k=d: D=O(log n/loglog n) • d<k<2d: D=O(logαn) α>1 • k=2d: D=O(nβ) 0<β<1 Suggest “two-step greedy routing”

  18. NoN-greedy routing Theorem(Manku, Naor, Wieder [STOC ‘04]) In d-dimensional meshes augmented with the d-dimensional harmonic distribution, with c long links per node, NoN-greedy routing performs in O(log2n/(c log c)) expected #steps. In particular: c = log n O(log n / loglog n) steps

  19. Local awareness (1)

  20. Local awareness (2) x Awareness(x)

  21. Indirect-greedy routing • Curent node x selects node y in awareness(x) whose long link is the closest to the target t; • Node x uses (Kleinberg) greedy routing to route in direction of y;

  22. Performances of Indirect-greedy routing Theorem(F., Gavoille, Paul [PODC ‘04]) In d-dimensional meshes augmented with the d-harmonic distribution, indirect-greedy routing with an awareness of O(log2n) bits per node performs in O(log1+1/dn) expected #steps. Eclecticism shrinks the world!

  23. KGR is better KGR Large #ID ID too far Awareness O(log n) is optimal Exp. #steps log2n log1+1/dn Size awareness log n logdn

  24. Conclusion

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