Improved Extended Regular Expression Matching

Abstract

An extended regular expression $R$ specifies a set of strings formed by characters from an alphabet combined with concatenation, union, intersection, complement, and star operators. Given an extended regular expression $R$ and a string $Q$, the extended regular expression matching problem is to decide if $Q$ matches any of the strings specified by $R$. Extended regular expression matching was introduced by Hopcroft and Ullman in the 1970s, who gave a simple dynamic programming solution using $O(n^{3}m)$ time and $O(n^{2}m)$ space, where $n$ is the length of $Q$ and $m$ is the length of $R$. The current state-of-the art solution, by Yamamoto and Miyazaki uses $O(\frac{n^{3}k+n^{2}m}{w}+n+m)$ time and $O(\frac{n^{2}k+nm}{w}+n+m)$ space, where $k$ is the number of negation and complement operators in $R$ and $w$ is the number of bits in a machine word. This roughly replaces the $m$ factor with $k$ in the dominant terms of both the space and time bounds of the classical Hopcroft and Ullman algorithm. In this paper, we present a new solution that solves extended regular expression matching in \[ O\left(n^ωk + \frac{n^2m}{\max(w/\log w, \log n)} + m\right) \] time and $O(\frac{n^2\log k}{w}+n+m)=O(n^2+m)$ space, where $\omega \approx 2.3716$ is the exponent of matrix multiplication. Essentially, this replaces the dominant $n^{3}k$ term with $n^{\omega }k$ in the time bound, while simultaneously improving the $n^{2}k$ term in the space to $O(n^2)$. Our results are based on a surprisingly simple combination of techniques and insights, including a compact representation to store and efficiently combine substring matches, a clustering technique for parse trees of extended regular expressions, and a new efficient combination of finite automaton simulation with our substring match representation to speed up the classic dynamic programming solution.