Faster Algorithms for String Matching Problems: Matching the Convolution Bound

In this paper we give a randomized O(n logn)-time algorithm for the string matching with don't cares problem. This improves the Fischer-Paterson bound 10] from 1974 and answers the open problem posed (among others) by Weiner 30] and Galil 11]. Using the same technique, we give an O(n logn)-time algorithm for other problems, including subset matching and tree pattern matching 15, 21, 9, 7, 17] and (general) approximate threshold matching 28, 17]. As this bound essentially matches the complexity of computing of the Fast Fourier Transform which is the only known technique for solving problems of this type, it is likely that the algorithms are in fact optimal. Additionally, the technique used for the threshold matching problem can be applied to the on-line version of this problem, in which we are allowed to preprocess the text and require to process the pattern in time sublinear in the text length. This result involves an interesting variant of the Karp-Rabin ngerprint method 22] in which hash functions are locality-sensitive 25], i.e. the probability of collision of two words depends on the distance between them.