Matching a Set of Strings with Variable Length Don't Cares

Given an alphabet A, a pattern p is a sequence (v l , . . . ,vm) of words from A* called keywords. We represent p as a single word vl@.. 9 @vm, where @ ~ A is a distinguished symbol called variable length don't care symbol. Pattern p is said to match a text t E A* if t = UoVlUa...um_xv,~u,~ for some u0 , . . . ,u ,~ E A' . In this paper we address the following problem: given a set P of patterns and a text t, test whether one of the patterns of P matches t. Quoting Fisher and Paterson in the concluding section of [10], "a good algorithm for this (problem) would have obvious practical applications". For instance, as it was reported by Manber and Baeza-Yate s [13], the DNA pat tern TATA often appears after the pat tern CAATCT within a variable length space. It may therefore be interesting to look for the general pat tern CAATCT@TATA. If we are given a set of such general patterns, it is desirable to have an algorithm that searches for all of them simultaneously instead of searching consecutively for each one. In this paper we propose an algorithm that solves the problem in t ime O(([t[ + [P])log IP[), where ]t I is the length of the text and [P[ is the total length of all keywords of P. Several variants of the problem have been considered in the literature 9 Matching set of strings with "unit length don't care symbols" that match any individual letter, was studied in [10, 15]. Bertossi a~d Logi [5] have proposed an efficient parallel algorithm for finding in a text the occurrences of a single pat tern with variable length don't-care symbols. Their algorithm has an O(log ]t[) running t ime on O([t[[P[/log [tl) processors. Our problem can also be viewed as testing membership of a word in a regular language of type '~ 9 i 9 i 9 9 i 9 tJi=lA u lA u2A .. Note that any 9 A u,~A . regular expression where the star operation only applies to the subexpression A (i.e. the union of all letters) can be reduced to the above form by distributing concatenation over union. An O(]t[[Z[/log [t I) solution for the case of a general regular expression E has been given by Myers [14]. The algorithm we propose here reads the text and the patterns from different tapes in the left-to-right fashion. The text is searched on-line, which means that the match is reported immediately after reading the shortest matched portion of the