Testing Connectedness of Images - Berman, Piotr; Murzabulatov, Meiram; Raskhodnikova, Sofya; Ristache, Dragos-Florian (Algorithmica, 2024)
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 ϵ 2 ) queries and adaptively with O ( 1 ϵ 3 / 2 log 1 ϵ ) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O ( 1 ϵ 2 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.