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 $O(\frac 1{\epsilon^2})$ queries and adaptively with $O(\frac{1}{\epsilon^{3/2}} \sqrt{\log\frac{1}{\epsilon}})$ queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity $O(\frac 1{\epsilon^2}\log \frac 1{\epsilon})$ and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make $\Omega(\frac 1\epsilon\log \frac 1\epsilon)$ queries.

翻译：我们研究了测试图像是否连通的算法。给定一个邻近参数 $\epsilon\in(0,1)$ 以及对以 $n\times n$ 布尔像素值矩阵表示的黑白图像的查询访问权限，一个（单侧错误）连通性测试器在图像连通时接受输入，并在图像与连通图像相差至少 $\epsilon$ 时以至少 2/3 的概率拒绝输入。我们证明了连通性可以通过非自适应查询 $O(\frac 1{\epsilon^2})$ 次以及自适应查询 $O(\frac{1}{\epsilon^{3/2}} \sqrt{\log\frac{1}{\epsilon}})$ 次进行测试。目前最佳的连通性测试器由 Berman、Raskhodnikova 和 Yaroslavtsev（STOC 2014）提出，其查询复杂度为 $O(\frac 1{\epsilon^2}\log \frac 1{\epsilon})$ 且为自适应算法。我们还证明了任何非自适应的单侧错误连通性测试器必须进行 $\Omega(\frac 1\epsilon\log \frac 1\epsilon)$ 次查询。