Lookup in Small Worlds -- A Survey --
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Lookup in Small Worlds -- A Survey --
Pierre Fraigniaud CNRS, U. Paris Sud
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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”
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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
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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
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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)
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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
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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?
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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
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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?
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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?
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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?
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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).
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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)
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Increasing the awareness
Neighbors-of-neighbors (NoN)
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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
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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)
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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”
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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
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Local awareness (1)
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Local awareness (2)
x
Awareness(x)
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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;
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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!
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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
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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