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Given samples from two distributions over an $n$-element set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in $n$, specifically, $O(n^{2/3}\epsilon^{-8/3}\log n)$, independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than $\max\{\epsilon^{4/3}n^{-1/3}/32, \epsilon n^{-1/2}/4\}$) or large (more than $\epsilon$) in $\ell_1$ distance. This result can be compared to the lower bound of $\Omega(n^{2/3}\epsilon^{-2/3})$ for this problem given by Valiant. Our algorithm has applications to the problem of testing whether a given Markov process is rapidly mixing. We present sublinear for several variants of this problem as well. A preliminary version of this paper appeared in the 41st Symposium on Foundations of Computer Science, 2000, Redondo Beach, C

Topics:
QA Mathematics

Publisher: arXiv.org

Year: 2010

OAI identifier:
oai:eprints.lse.ac.uk:31299

Provided by:
LSE Research Online

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