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.
@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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