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Pittsburgh, Pennsylvania, USA

Oct. 23, 2005 to Oct. 25, 2005

ISBN: 0-7695-2468-0

pp: 397-406

Avi Wigderson , Institute for Advanced Study, Princeton Univeristy

David Xiao , Princeton Univeristy

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/SFCS.2005.8

ABSTRACT

<p>In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter [1], in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is a generalization of Gillman?s and Lezaud?s analyses of the Ajtai-Komlos-Szemeredi sampler for realvalued functions [11, 21, 2].</p> <p>Derandomizing our sampler gives a few applications, yielding deterministic polynomial time algorithms for problems in which derandomizing independent sampling gives only quasi-polynomial time deterministic algorithms. The first (which was our original motivation) is to a polynomialtime derandomization of the Alon-Roichmantheorem [4, 20, 22]: given a group of size n, find O(log n) elements which generate it as an expander. This implies a second application efficiently constructing a randomness-optimal homomorphism tester, significantly improving the previous result of Shpilka and Wigderson [29]. A third application, which derandomizes a generalization of the set cover problem, is deferred to the full version of this paper.</p>

INDEX TERMS

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CITATION

Avi Wigderson,
David Xiao,
"A Randomness-Efficient Sampler for Matrix-valued Functions and Applications",

*FOCS*, 2005, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2005, pp. 397-406, doi:10.1109/SFCS.2005.8