On-line weighted pattern matching

A weighted sequence is a sequence of probability distributions over an alphabet of sizes sigma. Weighted sequences arise naturally in many applications. We study the problem of weighted pattern matching in which we are given a string pattern P of length m, a weight threshold 1/z, and a weighted text Xarriving on-line. We say that Poccurs in X at position i if the product of probabilities of the letters of P at positions i-m + 1, . . . , i in X is at least 1/z. We first discuss how to apply a known general scheme that transforms off-line pattern matching algorithms to on-line algorithms to obtain an on-line algorithm that requires O((sigma+ logz) logm) or O(sigma log(2) m) time per arriving position; with the space requirement however being O(m min(sigma, z)). Our main result is a new algorithm that processes each arriving position of X in O(z+ sigma) time using O(m + z) extra space. (C) 2019 Elsevier Inc. All rights reserved.