  We investigate the problem of testing the equivalence between two discrete
histograms. A {\em $k$-histogram} over $[n]$ is a probability distribution that
is piecewise constant over some set of $k$ intervals over $[n]$. Histograms
have been extensively studied in computer science and statistics. Given a set
of samples from two $k$-histogram distributions $p, q$ over $[n]$, we want to
distinguish (with high probability) between the cases that $p = q$ and
$\|p-q\|_1 \geq \epsilon$. The main contribution of this paper is a new
algorithm for this testing problem and a nearly matching information-theoretic
lower bound. Specifically, the sample complexity of our algorithm matches our
lower bound up to a logarithmic factor, improving on previous work by
polynomial factors in the relevant parameters. Our algorithmic approach applies
in a more general setting and yields improved sample upper bounds for testing
closeness of other structured distributions as well.
