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Efficient Matching of Some Fundamental Regular Expressions with Backreferences

Authors Taisei Nogami , Tachio Terauchi

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LIPIcs.MFCS.2025.81.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Pattern matching
Keywords
  • Regular expressions
  • Backreferences
  • Regex matching
  • NFA simulation
  • Suffix arrays
  • Right-maximal repeats

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Abstract

Regular expression matching is of practical importance due to its widespread use in real-world applications. In practical use, regular expressions are often used with real-world extensions. Accordingly, the matching problem of regular expressions with real-world extensions has been actively studied in recent years, yielding steady progress. However, backreference, a popular extension supported by most modern programming languages such as Java, Python, JavaScript and others in their standard libraries for string processing, is an exception to this positive trend. In fact, it is known that the matching problem of regular expressions with backreferences (rewbs) is theoretically hard and the existence of an asymptotically fast matching algorithm for arbitrary rewbs seems unlikely. Even among currently known partial solutions, the balance between efficiency and generality remains unsatisfactory. To bridge this gap, we present an efficient matching algorithm for rewbs of the form e_0 (e)_1 e_1 \1 e_2 where e_0, e, e_1, e_2 are pure regular expressions, which are fundamental and frequently used in practical applications. It runs in quadratic time with respect to the input string length, substantially improving the best-known cubic time complexity for these rewbs. Our algorithm combines ideas from both stringology and automata theory in a novel way. We leverage two techniques from automata theory, injection and summarization, to simultaneously examine matches whose backreferenced substrings are either a fixed right-maximal repeat or its extendable prefixes, which are concepts from stringology. By further utilizing a subtle property of extendable prefixes, our algorithm correctly decides the matching problem while achieving the quadratic-time complexity.

Cite As Get BibTex

Taisei Nogami and Tachio Terauchi. Efficient Matching of Some Fundamental Regular Expressions with Backreferences. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.81

Author Details

Taisei Nogami
  • Waseda University, Tokyo, Japan
Tachio Terauchi
  • Waseda University, Tokyo, Japan

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP25KJ2137, JP23K24826, and JP20K20625.

Supplementary Materials

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