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### Revision(s):

Revision #1 to TR15-160 | 29th January 2016 16:47

#### Are Few Bins Enough: Testing Histogram Distributions

Revision #1
Authors: Clement Canonne
Accepted on: 29th January 2016 16:47
Keywords:

Abstract:

A probability distribution over an ordered universe $[n]=\{1,\dots,n\}$ is said to be a $k$-histogram if it can be represented as a piecewise-constant function over at most $k$ contiguous intervals. We study the following question: given samples from an arbitrary distribution $D$ over $[n]$, one must decide whether $D$ is a $k$-histogram, or is far in $\ell_1$ distance from any such succinct representation. We obtain a sample and time-efficient algorithm for this problem, complemented by a nearly-matching information-theoretic lower bound on the number of samples required for this task. Our results significantly improve on the previous state-of-the-art, due to Indyk, Levi, and Rubinfeld (2012) and Canonne, Diakonikolas, Gouleakis, and Rubinfeld (2015).

Changes to previous version:

Added some discussion; corrected some typos, and updated the bibliography.

### Paper:

TR15-160 | 30th September 2015 04:05

#### Are Few Bins Enough: Testing Histogram Distributions

TR15-160
Authors: Clement Canonne
Publication: 30th September 2015 16:21
A probability distribution over an ordered universe $[n]=\{1,\dots,n\}$ is said to be a $k$-histogram if it can be represented as a piecewise-constant function over at most $k$ contiguous intervals. We study the following question: given samples from an arbitrary distribution $D$ over $[n]$, one must decide whether $D$ is a $k$-histogram, or is far in $\ell_1$ distance from any such succinct representation. We obtain a sample and time-efficient algorithm for this problem, complemented by a nearly-matching information-theoretic lower bound on the number of samples required for this task. Our results significantly improve on the previous state-of-the-art, due to Indyk, Levi, and Rubinfeld (2012) and Canonne, Diakonikolas, Gouleakis, and Rubinfeld (2015).