Optimal Algorithms for Testing Closeness of Discrete Distributions | Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms | Society for Industrial and Applied Mathematics

We study the question of closeness testing for two discrete distributions. More precisely, given samples from two distributions p and q over an n-element set, we wish to distinguish whether p = q versus p is at least ε-far from q, in either `1 or `2 distance. Batu et al [BFR00, BFR13] gave the first sub-linear time algorithms for these problems, which matched the lower bounds of [Val11] up to a logarithmic factor in n, and a polynomial factor of ε. In this work, we present simple testers for both the `1 and `2 settings, with sample complexity that is information-theoretically optimal, to constant factors, both in the dependence on n, and the dependence on ε; for the `1 testing problem we establish that the sample complexity is Θ(max{n/ε, n/ε}).