On-line weighted pattern matching

A weighted sequence is a sequence of probability distributions over an alphabet of size σ. 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 [Formula presented], and a weighted text X arriving on-line. We say that P occurs 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 [Formula presented]. 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((σ+log⁡z)log⁡m) or O(σlog ² ⁡m) time per arriving position; with the space requirement however being O(mmin⁡(σ,z)). Our main result is a new algorithm that processes each arriving position of X in O(z+σ) time using O(m+z) extra space.