Sublinear Algorithms for Gap Edit Distance

Elazar Goldenberg, Robert Krauthgamer, Barna Saha

2019 (modified: 08 Nov 2022)FOCS 2019Readers: Everyone

Abstract: 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(n2) time, and a more sophisticated algorithm runs in time O(n + t2) 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, Koucḱy 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 Õ(n/t + t 3 ). This time bound is sublinear roughly for all t in [ω(1), o(n 1/3 )], which was not known before. The best previous algorithms solve this problem in sublinear time only for t = ω(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-ε gap problem in time Õ(n/t 1-ε + t 3 ).

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