Faster Parallel String Matching via Larger Deterministic Samples

Abstract

Building on previous results of Breslauer, Galil, and Vishkin, we obtain for every p( m) = Ω(log log m) an optimal speedup parallel string matching algorithm that can preprocess a pattern P of length m in time O( p( m)) and can then find all occurrences of P in a text of an arbitrary length in time O(log log m/log p( m)). Letting p( m) = (log m) ϵ, for any ϵ > 0, we obtain a constant O(1/ϵ) search time. Letting p( m) = log log m we obtain a search time of O(log log m/log log log m). The first result improves the preprocessing time of Galil′s constant time algorithm. The second result improves the search time of Breslauer′s and Galil′s O(log log m) time algorithm. The main ingredient used to obtain this trade-off is a new algorithm for the computation of deterministic samples of superlogarithmic size in sublogarithmic time. This algorithm generalizes a similar algorithm recently obtained by Crochemore, Gąsieniec, Rytter, Muthukrishnan, and Ramesh. We also show that in the parallel comparison model, the search can be performed in constant time after O(log log m) preprocessing rounds using an optimal number of comparisons. This completely matches a lower bound of Breslauer and Galil.