Rikke Schjeldrup Jessen

arXiv2026

Improved Regular Expression Matching with Simple Backreferences

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.

cs.DS↗

arXiv2026

Improved Extended Regular Expression Matching

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 $O(n^3m)$ time and $O(n^2m)$ 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 $O(\frac{n^3k + n^2m}{w} + n + m)$ time and $O(\frac{n^2k + nm}{w} + n + m)$ 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 $ω\approx 2.3716$ is the exponent of matrix multiplication. Essentially, this replaces the dominant $n^3k$ term with $n^ωk$ in the time bound, while simultaneously improving the $n^2k$ 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.

cs.DS↗