TY - JOUR

T1 - Testing convexity of figures under the uniform distribution

AU - Berman, Piotr

AU - Murzabulatov, Meiram

AU - Raskhodnikova, Sofya

N1 - Funding Information:
This research was supported by the NSF; CCF-1422975/1832228. NSF CAREER; CCF-0845701 A preliminary version of this paper appeared in the proceedings of the 32nd International Symposium on Computational Geometry, SoCG 2016 [5]. The second and third authors were supported in part by NSF award CCF-1422975/1832228 and NSF CAREER award CCF-0845701.
Funding Information:
information This research was supported by the NSF; CCF-1422975/1832228 NSF CAREER; CCF-0845701

PY - 2019/5

Y1 - 2019/5

N2 - We consider the following basic geometric problem: Given 휖 ∈ (0, 1/2), a 2-dimensional black-and-white figure is ∊- far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊- far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(휖 -4/3 ) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the Ω(휖 −3∕2 ) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates 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.

AB - We consider the following basic geometric problem: Given 휖 ∈ (0, 1/2), a 2-dimensional black-and-white figure is ∊- far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊- far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(휖 -4/3 ) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the Ω(휖 −3∕2 ) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates 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.

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U2 - 10.1002/rsa.20797

DO - 10.1002/rsa.20797

M3 - Article

AN - SCOPUS:85053547054

VL - 54

SP - 413

EP - 443

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -