research-article

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties

Authors:

Deeparnab Chakrabarty,

C. SeshadhriAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 13, Issue 2

Article No.: 20, Pages 1 - 30

https://doi.org/10.1145/3039241

Published: 10 March 2017 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 article, we provide 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 over arbitrary ranges.

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 that 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 that 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 HKalemaj IRaskhodnikova S(2024)Isoperimetric inequalities for real‐valued functions with applications to monotonicity testingRandom Structures & Algorithms10.1002/rsa.2121165:1(191-219)Online publication date: 29-Feb-2024
https://doi.org/10.1002/rsa.21211
Black HChakrabarty DSeshadhri CSaha BServedio R(2023)Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an Õ(n√d) Monotonicity TesterProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585167(233-241)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585167
Black HChakrabarty DSeshadhri C(2023) A d 1/2+o(1) Monotonicity Tester for Boolean Functions on d-Dimensional Hypergrids* 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00110(1796-1821)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00110
Show More Cited By

Index Terms

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties
1. Theory of computation
  1. Design and analysis of algorithms
    1. Streaming, sublinear and near linear time algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 13, Issue 2

Special Issue on SODA'15 and Regular Papers

April 2017

316 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3040971

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2017 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 10 March 2017

Accepted: 01 December 2016

Revised: 01 December 2016

Received: 01 May 2015

Published in TALG Volume 13, Issue 2

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Cited By

Black HKalemaj IRaskhodnikova S(2024)Isoperimetric inequalities for real‐valued functions with applications to monotonicity testingRandom Structures & Algorithms10.1002/rsa.2121165:1(191-219)Online publication date: 29-Feb-2024
https://doi.org/10.1002/rsa.21211
Black HChakrabarty DSeshadhri CSaha BServedio R(2023)Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an Õ(n√d) Monotonicity TesterProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585167(233-241)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585167
Black HChakrabarty DSeshadhri C(2023) A d 1/2+o(1) Monotonicity Tester for Boolean Functions on d-Dimensional Hypergrids* 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00110(1796-1821)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00110
Pallavoor RRaskhodnikova SWaingarten E(2021)Approximating the distance to monotonicity of Boolean functionsRandom Structures & Algorithms10.1002/rsa.2102960:2(233-260)Online publication date: 24-Jun-2021
https://doi.org/10.1002/rsa.21029
Dixit KRaskhodnikova SThakurta AVarma N(2018)Erasure-Resilient Property TestingSIAM Journal on Computing10.1137/16M107566147:2(295-329)Online publication date: Jan-2018
https://doi.org/10.1137/16M1075661
Pallavoor RRaskhodnikova SVarma N(2017)Parameterized Property Testing of FunctionsACM Transactions on Computation Theory10.1145/31552969:4(1-19)Online publication date: 14-Dec-2017
https://dl.acm.org/doi/10.1145/3155296

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