Fine-Grained Complexity of Regular Expression Pattern Matching and Membership

Author Philipp Schepper

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Philipp Schepper
  • CISPA - Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Saarbrücken, Germany


I thank Karl Bringmann for the supervision during the research for my Master’s Thesis which this paper is based on and especially the pointer to Batch-OV which simplified the upper bounds extremely.

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Philipp Schepper. Fine-Grained Complexity of Regular Expression Pattern Matching and Membership. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of bounded depth in this work. For them a classification splitting the types in easy (strongly sub-quadratic) and hard (essentially quadratic time under SETH) is known. We take a very fine-grained look at the hard pattern types from this classification and show a dichotomy: few types allow super-poly-logarithmic improvements while the algorithms for the other pattern types can only be improved by a constant number of log-factors, assuming the Formula-SAT Hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Fine-Grained Complexity
  • Regular Expression
  • Pattern Matching
  • Dichotomy


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