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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$\eps$-far from $q$, in either $\ell_1$ or $\ell_2$ distance. Batu et al. gavethe first sub-linear time algorithms for these problems, which matched thelower bounds of Valiant up to a logarithmic factor in $n$, and a polynomialfactor of $\eps.$  In this work, we present simple (and new) testers for both the $\ell_1$ and$\ell_2$ settings, with sample complexity that is information-theoreticallyoptimal, to constant factors, both in the dependence on $n$, and the dependenceon $\eps$; for the $\ell_1$ testing problem we establish that the samplecomplexity is $\Theta(\max\{n^{2/3}/\eps^{4/3}, n^{1/2}/\eps^2 \}).$

Optimal Algorithms for Testing Closeness of Discrete Distributions