Faster Pattern Matching under Edit Distance : A Reduction to Dynamic Puzzle Matching and the Seaweed Monoid of Permutation Matrices

We consider the approximate pattern matching problem under the edit distance. Given a text T of length n, a pattern P of length m, and a threshold k, the task is to find the starting positions of all substrings of T that can be transformed to P with at most k edits. More than 20 years ago, Cole and Hariharan [SODA’98, J. Comput.’02] gave an $\mathcal{O}(n+k^{4}\cdot n/m)$ time algorithm for this classic problem, and this runtime has not been improved since.Here, we present an algorithm that runs in time $\mathcal{O}\left(n+ k^{3.5}\sqrt{\log m\log k}\cdot n/m\right)$, thus breaking through this longstanding barrier. In the case where $n^{1/4+\varepsilon}\leq k\leq n^{2/5-\varepsilon}$ for some arbitrarily small positive constant $\varepsilon$, our algorithm improves over the state-of-the-art by polynomial factors: it is polynomially faster than both the algorithm of Cole and Hariharan and the classic $\mathcal{O}(kn)$-time algorithm of Landau and Vishkin [STOC’86, J. Algorithms’89].We observe that the bottleneck case of the alternative $\mathcal{O}(n+k^4 \cdot n / m$-time algorithm of Charalampopoulos, Kociumaka, and Wellnitz [FOCS’20] is when the text and the pattern are (almost) periodic. Our new algorithm reduces this case to a new Dynamic Puzzle Matching problem, which we solve by building on tools developed by Tiskin [SODA’10, Algorithmica’15] for the so-called seaweed monoid of permutation matrices. Our algorithm relies only on a small set of primitive operations on strings and thus also applies to the fully-compressed setting (where text and pattern are given as straight-line programs) and to the dynamic setting (where we maintain a collection of strings under creation, splitting, and concatenation), improving over the state of the art.