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 graphsWatts & 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

uv

prob(uv) ≈ 1/dist(u,v)d

Exactly 1 long link per node

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)

Performances of Greedy Routing

t

xdistG(x,t)=m

B=ball radius m/2

long link

long link

O(log n) expect. #steps to enter B

O(log2n) expect. #steps to reach t from s

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?

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

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?

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?

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?

A decentralized algorithm for routing

Theorem (Lebhar, Schabanel [ICALP ’04])

There exists a distributed routing protocol that

1. Visits O(log2n) expected #nodes;2. Discovers routes of expected length

O(log n (loglog n)2).

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)

Increasing the awareness

Neighbors-of-neighbors (NoN)

Percolation theory

0 ≤ pi ≤ 1 with Σi 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

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<k<2: D=O(logαn) α>0• k>2: D=Ω(n)

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”

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

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(log2n) bits per node performs in O(log1+1/dn) expected #steps.

Eclecticism shrinks the world!

Awareness O(log n) is optimal

Size awareness

Exp. #steps

log2n

log n logdn

log1+1/dn

Large #ID

ID too far

KGR is betterKGR

ConclusionE(#steps) |

awareness|

Greedy (harm.)

Θ(log2n / c) c log n

Greedy (any) Ω(log2n / (c loglog n))

c log n

Decentralized O(log2n / log2c) c log n

NoN-greedy O(log2n / (c log c)) c2 log n

Indirect-gdy O(log1+1/dn / c1/d) log2n