TY - GEN
T1 - A randomness-efficient sampler for matrix-valued functions and applications
AU - Wigderson, Avi
AU - Xiao, David
PY - 2005
Y1 - 2005
N2 - 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 real-valued functions [11, 21, 2]. 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 polynomial-time derandomization of the Alon-Roichman theorem [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.
AB - 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 real-valued functions [11, 21, 2]. 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 polynomial-time derandomization of the Alon-Roichman theorem [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.
UR - http://www.scopus.com/inward/record.url?scp=33748623632&partnerID=8YFLogxK
U2 - 10.1109/SFCS.2005.8
DO - 10.1109/SFCS.2005.8
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:33748623632
SN - 0769524680
SN - 9780769524689
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 397
EP - 406
BT - Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
T2 - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Y2 - 23 October 2005 through 25 October 2005
ER -