Sublinear Algorithms for Gap Edit Distance

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(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 nearlinear time [Andoni, Krauthgamer and Onak, FOCS 2010], and a constant-factor approximation in subquadratic time [Chakrabarty, Das, Goldenberg, Koucký 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 timeÕ( 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−ǫ gap problem in timeÕ( n t 1−ǫ + t 3 ).Previous Work Batu et al.'s algorithm distinguishes t = n α vs f (t) = Ω(n) in O(n max{2α−1,α/2} ) time for any fixed α > 1 [BEK + 03]. Their approach crucially depends on f (t) = Ω(n) and cannot distinguish between (say) n 0.1 and n 0.99 . The best sublinear-time algorithm known for gap edit distance, by Andoni and Onak [AO09], distinguishes between t = n α vs f (t) = n β for β > α in time O(n 2+α−2β+o(1) ). For the quadratic gap problem, i.e., β = 2α, this time bound is O(n 2−3α+o(1) ), which becomes worse as t gets smaller (as discussed earlier). For example, when t = n 1/4 , the known algorithm is not sublinear, whereas ours runs in timeÕ(n 3/4 ). Presence of repeated patterns make the gap edit distance problem significantly difficult to approximate. When no repetition is allowed, the state-of-the-art sublinear-time algorithms of [AN10] 1 Throughout, the tilde notationÕ(·) andω(·) hide factors that are polylogarithmic in n.

Does preprocessing help in fast sequence comparisons?

“…Contrasting to [GKS19], we again get exact result and better query time bound for all regimes of k, even when we allow single string preprocessing.…”

“…Small edit distance For general strings, if their edit distance is at most k, we can compute it exactly with O(n log(n)) preprocessing and O(k 2 log(n)) query time. Contrast this result with [GKS19] where inÕ( n k + k 3 ) time, one can distinguish if edit distance is below k or above Θ(k 2 ).…”

Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

“…In recent years, further progress was achieved, mostly by exploiting adaptive queries, particularly by Goldenberg, Krauthgamer, and Saha [GKS19], and subsequently by Kociumaka and Saha [KS20], who improved over [AO12] when k is small. The running time Õ( n k c−1 +k 3 ) of [GKS19] had an undesirable cubic dependency on k, which was improved to quadratic in [KS20].…”

“…Indeed, non-adaptive algorithms often have the added advantage of simplicity, but they entail much less power. So far, the best results in the regime of small k came through carefully using adaptive queries [GKS19,KS20]. In this context, it seemed plausible that adaptivity would be crucial to improving beyond [AO12] for large k as well.…”

A Linear-Time $n^{0.4}$-Approximation for Longest Common Subsequence

“…These strong hardness results naturally bring up the question whether LCS or edit distance can be efficiently approximated (namely, whether an algorithm with truly subquadratic time Opn 2´ε q for any constant ε ą 0, can produce a good approximation in the worst-case). In the last two decades, significant progress has been made towards designing efficient approximation algorithms for edit distance [14,13,15,9,7,21,23,29,17]; the latest achievement is a constant-factor approximation in almost-linear 1 time [8].…”