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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We investigate algorithms for testing whether an image is connected. Given a proximity parameter ε ∈ (0,1) and query access to a black-and-white image represented by an n×n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ε-far from connected. We show that connectedness can be tested nonadaptively with O(1/ε²) queries and adaptively with O(1/ε^{3/2} √{log1/ε}) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O(1/ε² log 1/ε) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make Ω(1/ε log 1/ε) queries.

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova, and Dragos Ristache. Testing Connectedness of Images. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berman_et_al:LIPIcs.APPROX/RANDOM.2023.66, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya and Ristache, Dragos}, title = {{Testing Connectedness of Images}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {66:1--66:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.66}, URN = {urn:nbn:de:0030-drops-188918}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.66}, annote = {Keywords: Property testing, sublinear-algorithms, lower bounds, connectivity, graphs} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

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 does not. 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(1/epsilon) samples. In contrast, we prove that convexity can be tested with O(1/epsilon) queries by testers that can make queries of their choice while uniform testers for this property require Omega(1/epsilon^{5/4}) samples. Previously, the fastest known tester for convexity needed Theta(1/epsilon^{4/3}) queries.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. The Power and Limitations of Uniform Samples in Testing Properties of Figures. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.FSTTCS.2016.45, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{The Power and Limitations of Uniform Samples in Testing Properties of Figures}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.45}, URN = {urn:nbn:de:0030-drops-68803}, doi = {10.4230/LIPIcs.FSTTCS.2016.45}, annote = {Keywords: Property testing, randomized algorithms, being a half-plane, convexity} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies the desired property). Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published.
We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that it becomes a half-plane? a representation of a convex object? a representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error epsilon, after reading a number of pixels polynomial in 1/epsilon and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/epsilon. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.)
Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons P_{epsilon} such that (1) every convex image has a nearby polygon in P_{epsilon} and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in P_{epsilon}. This construction might be of independent interest.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Tolerant Testers of Image Properties. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 90:1-90:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.ICALP.2016.90, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{Tolerant Testers of Image Properties}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {90:1--90:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.90}, URN = {urn:nbn:de:0030-drops-61959}, doi = {10.4230/LIPIcs.ICALP.2016.90}, annote = {Keywords: Computational geometry, convexity, half-plane, connectedness, propertytesting, tolerant property testing} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

We consider the following basic geometric problem: Given epsilon in (0,1/2), a 2-dimensional figure that consists of a black object and a white background is epsilon-far from convex if it differs in at least an epsilon fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is epsilon-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 epsilon-far from convex.
We show that Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(epsilon^{-4/3}) and thus beats the Omega(epsilon^{-3/2}) sample lower bound for learning convex figures under the uniform distribution from the work of Schmeltz (Data Structures and Efficient Algorithms,1992). 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.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing Convexity of Figures Under the Uniform Distribution. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.SoCG.2016.17, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{Testing Convexity of Figures Under the Uniform Distribution}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.17}, URN = {urn:nbn:de:0030-drops-59094}, doi = {10.4230/LIPIcs.SoCG.2016.17}, annote = {Keywords: Convex sets, 2D geometry, randomized algorithms, property testing} }