New Algorithms for Regular Expression Matching

TL;DR

This paper presents new algorithms for regular expression matching that improve time complexity bounds using linear space, especially effective when the machine word size is at least logarithmic in the input size.

Contribution

The authors develop algorithms that achieve faster regular expression matching times with linear space, surpassing previous bounds, particularly for larger word sizes.

Findings

Improved time bounds for regular expression matching with linear space.

Algorithms are optimal for certain word size regimes.

Significant performance gains when machine word size exceeds logarithmic input size.

Abstract

In this paper we revisit the classical regular expression matching problem, namely, given a regular expression $R$ and a string $Q$, decide if $Q$ matches one of the strings specified by $R$. Let $m$ and $n$ be the length of $R$ and $Q$, respectively. On a standard unit-cost RAM with word length $w \geq \log n$, we show that the problem can be solved in $O(m)$ space with the following running times: \begin{equation*} \begin{cases} O(n\frac{m \log w}{w} + m \log w) & \text{if $m > w$} \\ O(n\log m + m\log m) & \text{if $\sqrt{w} < m \leq w$} \\ O(\min(n+ m^2, n\log m + m\log m)) & \text{if $m \leq \sqrt{w}$.} \end{cases} \end{equation*} This improves the best known time bound among algorithms using $O(m)$ space. Whenever $w \geq \log^2 n$ it improves all known time bounds regardless of how much space is used.