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 Result: The “six degrees of separation”

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

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

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

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

Performances of greedy routing Theorem (Kleinberg [STOC ’00]) Greedy routing performs in O(log 2 n) expected #steps in d-dimensional meshes augmented with d-harmonic distribution. Application: DHT “Symphony” (Manku, Bawa, Raghavan [USENIX ’03]) Can we improve this bound?

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(log 2 n/c) expected #steps. In particular: c = log n  O(log n) steps

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

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(log 2 n) expected #steps. Can we do better using other distributions?

Another bad news! Theorem (Aspnes, Diamadi, Shah [PODC’02]) Greedy routing in directed rings augmented with any distribution performs in Ω(log 2 n/loglog n) expected #steps. Probably true in undirected rings, and in higher dimensions… Is it the end of the game?

A decentralized algorithm for routing Theorem (Lebhar, Schabanel [ICALP ’04]) There exists a distributed routing protocol that 1.Visits O(log 2 n) expected #nodes; 2.Discovers routes of expected length O(log n (loglog n) 2 ).

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

Increasing the awareness Neighbors-of-neighbors (NoN)

Percolation theory 0 ≤ p i ≤ 1 with Σ i p i = 1 Kleinberg: for every node x, chose c edges (x,y i ) with prob{(x,y i ) is chosen} = p i Remark: deg(x) = c Percolation: for every edge (x,y i ), prob{(x,y i ) is in the network} = c p i prob{(x,y i ) is not in the network} = 1 - c p i Remark: E(deg(x)) = c

Diameter of percolation graphs Benjamini, 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 0 k>2: D=Ω(n)

Diameter of percolation graphs Coppersmith, 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 1 k=2d: D=O(n β ) 0<β<1 Suggest “two-step greedy routing”

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(log 2 n/(c log c)) expected #steps. In particular: c = log n  O(log n / loglog n) steps

Local awareness (1)

Local awareness (2) x Awareness(x)

Indirect-greedy routing 1)Curent node x selects node y in awareness(x) whose long link is the closest to the target t; 2)Node x uses (Kleinberg) greedy routing to route in direction of y;

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(log 2 n) bits per node performs in O(log 1+1/d n) expected #steps. Eclecticism shrinks the world!

Awareness O(log n) is optimal Size awareness Exp. #steps log 2 n log nlog d n log 1+1/d n Large #ID ID too far KGR is better KGR

Conclusion E(#steps)|awareness| Greedy (harm.)Θ(log 2 n / c)c log n Greedy (any)Ω(log 2 n / (c loglog n))c log n DecentralizedO(log 2 n / log 2 c)c log n NoN-greedyO(log 2 n / (c log c))c 2 log n Indirect-gdyO(log 1+1/d n / c 1/d )log 2 n