Average optimal string matching in packed strings

Abstract

In this paper we are concerned with the basic problem of string pattern matching: preprocess one or multiple fixed strings over alphabet σ so as to be able to efficiently search for all occurrences of the string(s) in a given text T of length n. In our model, we assume that text and patterns are tightly packed so that any single character occupies log σ bits and thus any sequence of k consecutive characters in the text or the pattern occupies exactly k log σ bits. We first show a data structure that requires O(m) words of space (more precisely O(mlogm) bits of space) where m is the total size of the patterns and answers to search queries in average-optimal O(n/y) time where y is the length of the shortest pattern (y = m in case of a single pattern). This first data structure, while optimal in time, still requires O(mlogm) bits of space, which might be too much considering that the patterns occupy only m log σ bits of space. We then show that our data structure can be compressed to only use O(m log σ) bits of space while achieving query time O(n(log σ m)ε /y), with ε any constant such that 0 < ε < 1. We finally show two other direct applications: average optimal pattern matching with worst-case guarantees and average optimal pattern matching with k differences. In the meantime we also show a slightly improved worst-case efficient multiple pattern matching algorithm. © 2013 Springer-Verlag.