An Efficient Pattern Matching Algorithm on a Subclass of Context Free Grammars

Abstract

There is a close relationship between formal language theory and data compression. Since 1990’s various types of grammar-based text compression algorithms have been introduced. Given an input string, a grammar-based text compression algorithm constructs a context-free grammar that only generates the string. An interesting and challenging problem is pattern matching on context-free grammars $\mathcal{P}$ of size m and $\mathcal{T}$ of size n, which are the descriptions of pattern string P of length M and text string T of length N, respectively. The goal is to solve the problem in time proportional only to m and n, not to M nor N. Kieffer et al. introduced a very practical grammar-based compression method called multilevel pattern matching code (MPM code). In this paper, we propose an efficient pattern matching algorithm which, given two MPM grammars $\mathcal{P}$ and $\mathcal{T}$, performs in O(mn2) time with O(mn) space. Our algorithm outperforms the previous best one by Miyazaki et al. which requires O(m2n2) time and O(mn) space.