  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. gave
the first sub-linear time algorithms for these problems, which matched the
lower bounds of Valiant up to a logarithmic factor in $n$, and a polynomial
factor 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-theoretically
optimal, to constant factors, both in the dependence on $n$, and the dependence
on $\eps$; for the $\ell_1$ testing problem we establish that the sample
complexity is $\Theta(\max\{n^{2/3}/\eps^{4/3}, n^{1/2}/\eps^2 \}).$
