Testing Connectedness of Images

We investigate algorithms for testing whether an image is connected. Given a proximity parameter \({\epsilon }\in (0,1)\) and query access to a black-and-white image represented by an \(n\times n\) 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 \({\epsilon }\)-far from connected. We show that connectedness can be tested nonadaptively with \(O\Big (\frac{1}{{\epsilon }^2}\Big )\) queries and adaptively with \(O\Big (\frac{1}{{\epsilon }^{3/2}} \sqrt{\log \frac{1}{{\epsilon }}}\Big )\) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity \(O\Big (\frac{1}{{\epsilon }^2}\log \frac{1}{{\epsilon }}\Big )\) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make \(\Omega \Big (\frac{1}{{\epsilon }}\log \frac{1}{{\epsilon }}\Big )\) queries.