Improved Regular Expression Matching with Simple Backreferences

2026-09-15

AI Summary

该论文针对含简单回溯引用的正则表达式匹配问题，提出了一种时间复杂度为O(n^2m)、空间复杂度为O(nm)的新算法，改进了现有解决方案。

Abstract

A regular expression with backreferences (rewb) specifies a set of strings formed by characters combined with concatenation, union, star operators, and backreferences. A backreference consists of a capturing group $(\cdot)_i$ and a reference $\backslash i$. The substring matched by the reference must match the substring matched by the corresponding capturing group. Given a rewb $R$ and a string $Q$, the rewb matching problem is to decide whether $Q$ is one of the strings specified by $R$. In full generality, rewb matching is NP-complete, but efficient solutions exist for various subclasses. In the paper, we focus on rewb containing a single capturing group and $k$ references. For this class, Uezato~[CPM 2026] gave an $O((k n^2 m^2)$ time and $O(n^2m^2)$ space algorithm, where $m$ is the length of the regular expression $R$ and $n$ is the length of the string $Q$. For the special case of $k=1$, Nogami and Terauchi~[MFCS 2025] gave an $O(n^2m^2)$ time and $O(n+ m^2)$ space algorithm. On the other hand, Nogami, Nakamura, and Terauchi~[arXiv 2026] gave a conditional lower bound, showing that we cannot solve the problem in $O(n^{2-ε} \mathrm{poly}(m))$ for any $ε> 0$ assuming the orthogonal vector hypothesis. Our main result is a new algorithm that runs in $O(n^2m)$ time and uses $O(nm)$ space. This improves the above results (by a factor of $km$ and $m$, respectively) and the former's space bound (by a factor of $nm$). We also show how to extend our algorithm to handle a slightly more general class of ordered and single-nested rewbs.

Problem

• regular expression
• backreferences
• matching problem
• time complexity
• space complexity

Innovation

• Regular Expression
• Backreferences
• Algorithm Improvement
• Time Complexity
• Space Complexity