Approximate online pattern matching in sublinear time

We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern P of length m and a text T of length n over some alphabet Σ, and a positive integer k. The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position t in {1,...,n}, let kt be the smallest edit distance between P and any substring of T ending at t. In this paper we give a constant factor approximation to the sequence k1,k2,...,kn. We consider both offline and online settings. In the offline setting, where both P and T are available, we present an algorithm that for all t in {1,...,n}, computes the value of kt approximately within a constant factor. The worst case running time of our algorithm is Õ(nm³/⁴). In the online setting, we are given P and then T arrives one symbol at a time. We design an algorithm that upon arrival of the t-th symbol of T computes kt approximately within O(1)multiplicative factor and m⁸/⁹-additive error. Our algorithm takes Õ(m¹−(7/⁵⁴⁾) amortized time per symbol arrival and takes Õ(m¹−(1/⁵⁴⁾) additional space apart from storing the pattern P. Both of our algorithms are randomized and produce correct answer with high probability. To the best of our knowledge this is the first algorithm that takes worst-case sublinear (in the length of the pattern) time and sublinear extra space for the online approximate pattern matching problem. To get our result we build on the technique of Chakraborty, Das, Goldenberg, Koucký and Saks [FOCS'18] for computing a constant factor approximation of edit distance in sub-quadratic time.