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

Near-Optimal Closeness Testing of Discrete Histogram Distributions