Testing convexity of figures under the uniform distribution

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

We consider the following basic geometric problem: Given ε ∈ (0, 1/2), a 2-dimensional figure that consists of a black object and a white background is ε-far from convex if it differs in at least an ε fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ε-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity. Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is ε-far from convex. We show that Θ(ε-4/3) uniform samples are necessary and sufficient for detecting a violation of convexity in an ε-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(ε-4/3) and thus beats the Ω(ε-3/2) sample lower bound for learning convex figures under the uniform distribution from [26]. It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

Original languageEnglish (US)
Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
EditorsSandor Fekete, Anna Lubiw
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages17.1-17.15
ISBN (Electronic)9783959770095
DOIs
StatePublished - Jun 1 2016
Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
Duration: Jun 14 2016Jun 17 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume51
ISSN (Print)1868-8969

Other

Other32nd International Symposium on Computational Geometry, SoCG 2016
CountryUnited States
CityBoston
Period6/14/166/17/16

All Science Journal Classification (ASJC) codes

  • Software

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