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Matching Regular-Typed Pattern Languages: Quadratic-Time Algorithms

Author Yuya Uezato

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LIPIcs.CPM.2026.11.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Pattern matching
Keywords
  • Pattern languages
  • Regular expressions
  • String algorithms

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Abstract

Pattern languages (PAT) are a class of languages generated by expressions called patterns that may contain variables. In a pattern, each variable can be instantiated with an arbitrary string. Typed pattern languages extend PAT by associating a type (constraint) with each variable that restricts the domain of allowed substitutions. In this paper, we study regular-typed PAT (PATwRT), where all types are represented either by a regular expression or by an ε-NFA. We consider the PATwRT matching problem for patterns with a single repeated variable of the form P = α₁ β α₂ β ⋯ β α_K. We present simple algorithms whose running time is linear in K and quadratic in the input length N, with polynomial dependence on the sizes of the type representations. Our results extend previous quadratic-time work in two directions: (1) the quadratic-time algorithm for untyped PAT of Fernau et al. (STACS 2015), and (2) the quadratic-time algorithm for the restricted PATwRT K = 3, i.e., α₁ β α₂ β α₃ of Nogami and Terauchi (MFCS 2025).

Cite As Get BibTex

Yuya Uezato. Matching Regular-Typed Pattern Languages: Quadratic-Time Algorithms. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.11

Author Details

Yuya Uezato
  • CyberAgent, Inc., Tokyo, Japan
  • National Institute of Informatics, Tokyo, Japan

Funding

This work was supported by JST, CREST Grant Number JPMJCR21M3.

Acknowledgements

The author is deeply grateful to Soh Kumabe for fruitful discussions during the early stages of this research and for providing valuable comments. The author also thanks the anonymous reviewers for their insightful feedback, which significantly improved the presentation of this paper. In particular, one of the reviewers kindly suggested using an O(n^{ω})-time algorithm for computing the reflexive transitive closure of a Boolean matrix, which improved the running time of our algorithm for dense ε-NFAs.

References

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