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The Power and Limitations of Uniform Samples in Testing Properties of Figures

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Abstract

We investigate testing of properties of 2-dimensional figures that consist of a black object on a white background. Given a parameter \({\epsilon }\in (0,1/2)\), a tester for a specified property has to accept with probability at least 2/3 if the input figure satisfies the property and reject with probability at least 2/3 if it is \({\epsilon }\)-far from satisfying the property. In general, property testers can query the color of any point in the input figure. We study the power of testers that get access only to uniform samples from the input figure. We show that for the property of being a half-plane, the uniform testers are as powerful as general testers: they require only \(O({\epsilon }^{-1})\) samples. In contrast, we prove that convexity can be tested with \(O({\epsilon }^{-1})\) queries by testers that can make queries of their choice while uniform testers for this property require \(\varOmega ({\epsilon }^{-5/4})\) samples. Previously, the fastest known tester for convexity needed \(\varTheta ({\epsilon }^{-4/3})\) queries.

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Notes

  1. If \(\delta \) is not specified, it is assumed to be 1/3. By standard arguments, the error probability can be reduced from 1/3 to an arbitrarily small \(\delta \) by running the tester \(O(\log 1/\delta )\) times.

  2. For any nontrivial property, including being a half-plane, \(\varOmega ({\epsilon }^{-1})\) is an easy lower bound on the complexity of an \({\epsilon }\)-tester.

  3. For the two properties we consider (being a half-plane and convexity), we assume w.l.o.g. that the input figure U has unit area. If it is not the case, U can be rescaled. Thus, the area of a region corresponds to the probability of sampling from it under the uniform distribution.

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Authors and Affiliations

  1. Pennsylvania State University, University Park, USA

    Piotr Berman & Meiram Murzabulatov

  2. Boston University, Boston, USA

    Sofya Raskhodnikova

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  1. Piotr Berman
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  2. Meiram Murzabulatov
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Correspondence to Sofya Raskhodnikova.

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A preliminary version of this paper appeared in the proceedings of the 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, 2016 [10].

M. Murzabulatov and S. Raskhodnikova were supported by NSF award CCF-1422975.

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Berman, P., Murzabulatov, M. & Raskhodnikova, S. The Power and Limitations of Uniform Samples in Testing Properties of Figures. Algorithmica 81, 1247–1266 (2019). https://doi.org/10.1007/s00453-018-0467-9

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