research-article

Property testing on product distributions: optimal testers for bounded derivative properties

Authors:

Deeparnab Chakrabarty,

Madhav Jha, and

C. SeshadhriAuthors Info & Claims

SODA '15: Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms

January 2015

Pages 1809 - 1828

Published: 04 January 2015 Publication History

Abstract

The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions.

In this paper, we give simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing.

We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties, which relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best known result.

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Cited By

Black HChakrabarty DSeshadhri CCzumaj A(2018)A o(d) · polylog n monotonicity tester for boolean functions over the hypergrid [n]Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175444(2133-2151)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175444

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cover image ACM Other conferences

SODA '15: Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms

January 2015

2056 pages

Program Chair:
Piotr Indyk
Massachusetts Institute of Technology

Sponsors

SIAM: Society for Industrial and Applied Mathematics

In-Cooperation

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 04 January 2015

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SODA '15

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SIAM

SODA '15: ACM SIAM Symposium on Discrete Algorithms

January 4 - 6, 2015

California, San Diego

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SODA '15 Paper Acceptance Rate 137 of 495 submissions, 28%;

Overall Acceptance Rate 411 of 1,322 submissions, 31%

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Cited By

Black HChakrabarty DSeshadhri CCzumaj A(2018)A o(d) · polylog n monotonicity tester for boolean functions over the hypergrid [n]Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175444(2133-2151)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175444

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