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In the [[mathematics|mathematical]] discipline of [[graph theory]], the '''expander walk sampling theorem''' states that [[Sampling (statistics)|sampling]] [[vertex (graph theory)|vertices]] in an [[expander graph]] by doing a [[Random walk#Random walk on graphs|random walk]] is almost as good as sampling the vertices [[statistical independence|independently]] from a [[uniform distribution (discrete)|uniform distribution]].
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The earliest version of this theorem is due to {{harvtxt|Ajtai|Komlós|Szemerédi|1987}}, and the more general version is typically attributed to {{harvtxt|Gillman|1998}}.
 
==Statement==
Let <math>G = (V, E)</math> be an expander graph with [[Expander graph#Spectral expansion|normalized second-largest eigenvalue]] <math>\lambda</math>. Let <math>n</math> denote the number of vertices in <math>G</math>. Let <math>f : V \rightarrow [0, 1]</math> be a function on the vertices of <math>G</math>. Let <math>\mu = E[f]</math> denote the true mean of <math>f</math>, i.e. <math>\mu = \frac{1}{n} \sum_{v \in V} f(v)</math>.  Then, if we let <math>Y_0, Y_1, \ldots, Y_k</math> denote the vertices encountered in a <math>k</math>-step random walk on <math>G</math> starting at a random vertex <math>Y_0</math>, we have the following for all <math>\gamma > 0</math>:
 
:<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu > \gamma\right] \leq e^{-\Omega (\gamma^2 (1-\lambda) k)}.</math>
 
Here the <math>\Omega</math> hides an absolute constant <math>\geq 1/10</math>. An identical bound holds in the other direction:
 
:<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu < -\gamma\right] \leq e^{-\Omega (\gamma^2 (1-\lambda) k)}.</math>
 
==Uses==
This theorem is useful in randomness reduction in the study of [[derandomization]].  Sampling from an expander walk is an example of a randomness-efficient [[sample (statistics)|sampler]]. Note that the number of [[bit]]s used in sampling <math>k</math> independent samples from <math>f</math> is <math>k \log n</math>, whereas if we sample from an infinite family of constant-degree expanders this costs only <math>\log n + O(k)</math>.  Such families exist and are efficiently constructible, e.g. the [[Ramanujan graph]]s of [[Alexander Lubotzky|Lubotzky]]-Phillips-Sarnak.
 
==Notes==
{{reflist}}
 
==References==
{{refbegin}}
* {{citation
| first1=M. | last1=Ajtai
| first2=J. | last2=Komlós
| first3=E. | last3=Szemerédi
| title=Deterministic simulation in LOGSPACE
| booktitle=Proceedings of the nineteenth annual ACM symposium on Theory of computing
| pages=132–140
| year=1987
| work=ACM
| doi=10.1145/28395.28410
}}
* {{citation
| first=D. | last=Gillman
| title=A Chernoff Bound for Random Walks on Expander Graphs
| journal=SIAM Journal on Computing
| volume=27
| pages=1203–1220
| year=1998
| publisher=Society for Industrial and Applied Mathematics
| doi=10.1137/S0097539794268765
| issue=4,
}}
{{refend}}
 
==External links==
* Proofs of the expander walk sampling theorem. [http://citeseer.ist.psu.edu/gillman98chernoff.html] [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.aoap/1028903453]
 
[[Category:Sampling (statistics)]]

Latest revision as of 06:26, 2 January 2015

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