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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}}.
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| ==Statement==
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| 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>:
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| :<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> | |
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| Here the <math>\Omega</math> hides an absolute constant <math>\geq 1/10</math>. An identical bound holds in the other direction:
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| :<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>
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| ==Uses==
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| 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.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| * {{citation
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| | first1=M. | last1=Ajtai
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| | first2=J. | last2=Komlós
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| | first3=E. | last3=Szemerédi
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| | title=Deterministic simulation in LOGSPACE
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| | booktitle=Proceedings of the nineteenth annual ACM symposium on Theory of computing
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| | pages=132–140
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| | year=1987
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| | work=ACM
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| | doi=10.1145/28395.28410
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| }}
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| * {{citation
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| | first=D. | last=Gillman
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| | title=A Chernoff Bound for Random Walks on Expander Graphs
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| | journal=SIAM Journal on Computing
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| | volume=27
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| | pages=1203–1220
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| | year=1998
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| | publisher=Society for Industrial and Applied Mathematics
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| | doi=10.1137/S0097539794268765
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| | issue=4,
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| }}
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| {{refend}}
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| ==External links==
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| * 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]
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| [[Category:Sampling (statistics)]]
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I am Oscar and I totally dig that title. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for quite a while. South Dakota is where me and my husband reside. My working day occupation is a librarian.
Look at my web page ... sex-porn-tube.ch