Sublinear Algorithms for Gap Edit Distance

The edit distance is a way of quantifying how similar two strings are to one another by counting the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. A simple dynamic programming computes the edit distance between two strings of length $n$ in $O(n^2)$ time, and a more sophisticated algorithm runs in time $O(n+t^2)$ when the edit distance is $t$ [Landau, Myers and Schmidt, SICOMP 1998]. In pursuit of obtaining faster running time, the last couple of decades have seen a flurry of research on approximating edit distance, including polylogarithmic approximation in near-linear time [Andoni, Krauthgamer and Onak, FOCS 2010], and a constant-factor approximation in subquadratic time [Chakrabarty, Das, Goldenberg, Kouck\'y and Saks, FOCS 2018]. We study sublinear-time algorithms for small edit distance, which was investigated extensively because of its numerous applications. Our main result is an algorithm for distinguishing whether the edit distance is at most $t$ or at least $t^2$ (the quadratic gap problem) in time $\tilde{O}(\frac{n}{t}+t^3)$. This time bound is sublinear roughly for all $t$ in $[\omega(1), o(n^{1/3})]$, which was not known before. The best previous algorithms solve this problem in sublinear time only for $t=\omega(n^{1/3})$ [Andoni and Onak, STOC 2009]. Our algorithm is based on a new approach that adaptively switches between uniform sampling and reading contiguous blocks of the input strings. In contrast, all previous algorithms choose which coordinates to query non-adaptively. Moreover, it can be extended to solve the $t$ vs $t^{2-\epsilon}$ gap problem in time $\tilde{O}(\frac{n}{t^{1-\epsilon}}+t^3)$.