Document Open Access Logo

Linear Time Subsequence and Supersequence Regex Matching

Authors Antoine Amarilli , Florin Manea , Tina Ringleb , Markus L. Schmid

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2025.9.pdf
  • Filesize: 0.78 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
Keywords
  • subsequence
  • supersequence
  • regular language
  • regular expression
  • automata

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

It is well-known that checking whether a given string w matches a given regular expression r can be done in quadratic time O(|w|⋅ |r|) and that this cannot be improved to a truly subquadratic running time of O((|w|⋅ |r|)^{1-ε}) assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether w has a subsequence that matches r, and show that regex matching in this sense can be solved in linear time O(|w| + |r|). Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of w that matches r can be solved in O(|w|⋅ |r|), i. e., asymptotically no worse than classical regex matching; and we show that O(|w| + |r|) is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.

Cite As Get BibTex

Antoine Amarilli, Florin Manea, Tina Ringleb, and Markus L. Schmid. Linear Time Subsequence and Supersequence Regex Matching. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.9

Author Details

Antoine Amarilli
  • Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France
Florin Manea
  • Institute for Computer Science and CIDAS, University of Göttingen, Germany
Tina Ringleb
  • Institute for Computer Science, University of Göttingen, Germany
Markus L. Schmid
  • Humboldt-Universität zu Berlin, Germany

Funding

  • Amarilli, Antoine: Partially supported by the ANR project EQUUS ANR-19-CE48-0019 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 431183758.
  • Manea, Florin: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Heisenberg grant 466789228.
  • Schmid, Markus L.: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) – project number 522576760 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 522576760).

References

  1. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In FOCS, 2015. URL: https://doi.org/10.1109/FOCS.2015.14.
  2. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In ICALP, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_4.
  3. Parosh Aziz Abdulla, Ahmed Bouajjani, and Bengt Jonsson. On-the-fly analysis of systems with unbounded, lossy fifo channels. In CAV, 1998. URL: https://doi.org/10.1007/BFB0028754.
  4. Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea, and Dirk Nowotka. k-Universality of regular languages. In ISAAC, volume 283 of LIPIcs, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.4.
  5. Antoine Amarilli, Florin Manea, Tina Ringleb, and Markus L. Schmid. Linear time subsequence and supersequence regex matching. CoRR, abs/2504.16288, 2025. URL: https://doi.org/10.48550/arXiv.2504.16288.
  6. Renzo Angles, Marcelo Arenas, Pablo Barceló, Aidan Hogan, Juan L. Reutter, and Domagoj Vrgoc. Foundations of modern query languages for graph databases. ACM Comput. Surv., 50(5), 2017. URL: https://doi.org/10.1145/3104031.
  7. Alexander Artikis, Alessandro Margara, Martín Ugarte, Stijn Vansummeren, and Matthias Weidlich. Complex event recognition languages: Tutorial. In DEBS, 2017. URL: https://doi.org/10.1145/3093742.3095106.
  8. Georg Bachmeier, Michael Luttenberger, and Maximilian Schlund. Finite automata for the sub-and superword closure of CFLs: Descriptional and computational complexity. In ICALP, 2015. URL: https://doi.org/10.1007/978-3-319-15579-1_37.
  9. Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? In FOCS, 2016. URL: https://doi.org/10.1109/FOCS.2016.56.
  10. Lasse Bergroth, Harri Hakonen, and Timo Raita. A survey of longest common subsequence algorithms. In SPIRE, 2000. URL: https://doi.org/10.1109/SPIRE.2000.878178.
  11. Karl Bringmann and Bhaskar Ray Chaudhury. Sketching, streaming, and fine-grained complexity of (weighted) LCS. In FSTTCS, LIPIcs, 2018. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2018.40.
  12. Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen. The NFA acceptance hypothesis: Non-combinatorial and dynamic lower bounds. In ITCS, 2024. URL: https://doi.org/10.4230/LIPICS.ITCS.2024.22.
  13. Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 79-97, 2015. URL: https://doi.org/10.1109/FOCS.2015.15.
  14. Karl Bringmann and Marvin Künnemann. Multivariate fine-grained complexity of longest common subsequence. In SODA, 2018. URL: https://doi.org/10.1137/1.9781611975031.79.
  15. Sam Buss and Michael Soltys. Unshuffling a square is NP-hard. J. Comput. Syst. Sci., 80(4), 2014. URL: https://doi.org/10.1016/j.jcss.2013.11.002.
  16. Nicola Cotumaccio, Giovanna D’Agostino, Alberto Policriti, and Nicola Prezza. Co-lexicographically ordering automata and regular languages - Part I. J. ACM, 70(4), 2023. URL: https://doi.org/10.1145/3607471.
  17. Maxime Crochemore, Christophe Hancart, and Thierry Lecroq. Algorithms on strings. Cambridge University Press, 2007. Google Scholar
  18. Maxime Crochemore, Borivoj Melichar, and Zdenek Tronícek. Directed acyclic subsequence graph - overview. J. Discrete Algorithms, 1(3-4), 2003. URL: https://doi.org/10.1016/S1570-8667(03)00029-7.
  19. Joel D. Day, Pamela Fleischmann, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. The edit distance to k-subsequence universality. In STACS, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.25.
  20. Ronald Fagin, Benny Kimelfeld, Frederick Reiss, and Stijn Vansummeren. Document spanners: A formal approach to information extraction. J. ACM, 62(2), 2015. URL: https://doi.org/10.1145/2699442.
  21. Szilárd Zsolt Fazekas, Tore Koß, Florin Manea, Robert Mercas, and Timo Specht. Subsequence matching and analysis problems for formal languages. In ISAAC, volume 322 of LIPIcs, 2024. URL: https://doi.org/10.4230/LIPICS.ISAAC.2024.28.
  22. Michael L. Fredman and Dan E. Willard. BLASTING through the information theoretic barrier with FUSION TREES. In STOC, 1990. URL: https://doi.org/10.1145/100216.100217.
  23. Dominik D. Freydenberger, Pawel Gawrychowski, Juhani Karhumäki, Florin Manea, and Wojciech Rytter. Testing k-binomial equivalence. In Multidisciplinary Creativity, a collection of papers dedicated to G. Păun 65th birthday, 2015. available in CoRR abs/1509.00622. Google Scholar
  24. Jeffrey E. F. Friedl. Mastering regular expressions - Understand your data and be more productive: for Perl, PHP, Java, .NET, Ruby, and more. O'Reilly, 2006. Google Scholar
  25. André Frochaux and Sarah Kleest-Meißner. Puzzling over subsequence-query extensions: Disjunction and generalised gaps. In AMW, 2023. URL: https://ceur-ws.org/Vol-3409/paper3.pdf.
  26. Moses Ganardi, Irmak Sağlam, and Georg Zetzsche. Directed regular and context-free languages. In STACS, 2024. Google Scholar
  27. Pawel Gawrychowski, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Efficiently testing Simon’s congruence. In STACS, volume 187 of LIPIcs, 2021. URL: https://doi.org/10.4230/LIPICS.STACS.2021.34.
  28. Nikos Giatrakos, Elias Alevizos, Alexander Artikis, Antonios Deligiannakis, and Minos N. Garofalakis. Complex event recognition in the big data era: A survey. VLDB J., 29(1), 2020. URL: https://doi.org/10.1007/s00778-019-00557-w.
  29. Jean Goubault-Larrecq, Simon Halfon, Prateek Karandikar, K Narayan Kumar, and Philippe Schnoebelen. The ideal approach to computing closed subsets in well-quasi-orderings. Well-Quasi Orders in Computation, Logic, Language and Reasoning: A Unifying Concept of Proof Theory, Automata Theory, Formal Languages and Descriptive Set Theory, 2020. Google Scholar
  30. Étienne Grandjean and Louis Jachiet. Which arithmetic operations can be performed in constant time in the RAM model with addition?, 2023. URL: https://arxiv.org/abs/2206.13851.
  31. Alejandro Grez, Cristian Riveros, Martín Ugarte, and Stijn Vansummeren. A formal framework for complex event recognition. ACM Trans. Database Syst., 46(4), 2021. URL: https://doi.org/10.1145/3485463.
  32. Simon Halfon, Philippe Schnoebelen, and Georg Zetzsche. Decidability, complexity, and expressiveness of first-order logic over the subword ordering. In LICS, 2017. URL: https://doi.org/10.1109/LICS.2017.8005141.
  33. Daniel S. Hirschberg. Algorithms for the longest common subsequence problem. J. ACM, 24(4), 1977. URL: https://doi.org/10.1145/322033.322044.
  34. John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 2001. Google Scholar
  35. James W. Hunt and Thomas G. Szymanski. A fast algorithm for computing longest subsequences. Communications of the ACM, 20(5), 1977. URL: https://doi.org/10.1145/359581.359603.
  36. Prateek Karandikar and Philippe Schnoebelen. The height of piecewise-testable languages and the complexity of the logic of subwords. Log. Methods Comput. Sci., 15(2), 2019. URL: https://doi.org/10.23638/LMCS-15(2:6)2019.
  37. S.C. Kleene. Representation of events in nerve nets and finite automata. In Automata Studies, volume 34 of Annals of Mathematics Studies, pages 3-41. Princeton University Press, 1956. Google Scholar
  38. Sarah Kleest-Meißner, Rebecca Sattler, Markus L. Schmid, Nicole Schweikardt, and Matthias Weidlich. Discovering event queries from traces: Laying foundations for subsequence-queries with wildcards and gap-size constraints. In ICDT, 2022. Google Scholar
  39. Sarah Kleest-Meißner, Rebecca Sattler, Markus L. Schmid, Nicole Schweikardt, and Matthias Weidlich. Discovering multi-dimensional subsequence queries from traces - from theory to practice. In Datenbanksysteme für Business, Technologie und Web (BTW 2023), 20. Fachtagung des GI-Fachbereichs ,,Datenbanken und Informationssysteme" (DBIS), 2023. URL: https://doi.org/10.18420/BTW2023-24.
  40. Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Combinatorial algorithms for subsequence matching: A survey. In NCMA, volume 367 of EPTCS, 2022. URL: https://doi.org/10.4204/EPTCS.367.2.
  41. Sailesh Kumar, Balakrishnan Chandrasekaran, Jonathan S. Turner, and George Varghese. Curing regular expressions matching algorithms from insomnia, amnesia, and acalculia. In ANCS, 2007. URL: https://doi.org/10.1145/1323548.1323574.
  42. Dietrich Kuske. The subtrace order and counting first-order logic. In CSR, volume 12159 of LNCS, 2020. URL: https://doi.org/10.1007/978-3-030-50026-9_21.
  43. Dietrich Kuske and Georg Zetzsche. Languages ordered by the subword order. In FOSSACS, volume 11425 of LNCS, 2019. URL: https://doi.org/10.1007/978-3-030-17127-8_20.
  44. Alex X. Liu and Eric Norige. A de-compositional approach to regular expression matching for network security. IEEE/ACM Trans. Netw., 27(6), 2019. URL: https://doi.org/10.1109/TNET.2019.2941920.
  45. David Maier. The complexity of some problems on subsequences and supersequences. J. ACM, 25(2), 1978. URL: https://doi.org/10.1145/322063.322075.
  46. Alexandru Mateescu, Arto Salomaa, and Sheng Yu. Subword histories and Parikh matrices. J. Comput. Syst. Sci., 68(1):1-21, 2004. URL: https://doi.org/10.1016/J.JCSS.2003.04.001.
  47. Alexander Okhotin. On the state complexity of scattered substrings and superstrings. Fundamenta Informaticae, 99(3), 2010. URL: https://doi.org/10.3233/FI-2010-252.
  48. M. Praveen, Philippe Schnoebelen, Julien Veron, and Isa Vialard. On the piecewise complexity of words and periodic words. In SOFSEM, 2024. URL: https://doi.org/10.1007/978-3-031-52113-3_32.
  49. Steven Purtzel and Matthias Weidlich. Suse: Summary selection for regular expression subsequence aggregation over streams. In SIGMOD, 2025. To appear. Google Scholar
  50. William E. Riddle. An approach to software system modelling and analysis. Comput. Lang., 4(1), 1979. URL: https://doi.org/10.1016/0096-0551(79)90009-2.
  51. Michel Rigo and Pavel Salimov. Another generalization of abelian equivalence: Binomial complexity of infinite words. Theor. Comput. Sci., 601, 2015. URL: https://doi.org/10.1016/J.TCS.2015.07.025.
  52. Arto Salomaa. Connections between subwords and certain matrix mappings. Theoret. Comput. Sci., 340(2):188-203, 2005. URL: https://doi.org/10.1016/J.TCS.2005.03.024.
  53. Philippe Schnoebelen and Julien Veron. On arch factorization and subword universality for words and compressed words. In WORDS, 2023. URL: https://doi.org/10.1007/978-3-031-33180-0_21.
  54. Shinnosuke Seki. Absoluteness of subword inequality is undecidable. Theor. Comput. Sci., 418, 2012. URL: https://doi.org/10.1016/J.TCS.2011.10.017.
  55. Alan C. Shaw. Software descriptions with flow expressions. IEEE Trans. Software Eng., 4(3), 1978. URL: https://doi.org/10.1109/TSE.1978.231501.
  56. Imre Simon. Hierarchies of events with dot-depth one. PhD thesis, University of Waterloo, 1972. Google Scholar
  57. Imre Simon. Piecewise testable events. In Automata Theory and Formal Languages, 2nd GI Conference, volume 33 of LNCS, 1975. URL: https://doi.org/10.1007/3-540-07407-4_23.
  58. Imre Simon. Words distinguished by their subwords (Extended abstract). In WORDS, volume 27 of TUCS General Publication, 2003. Google Scholar
  59. Peter Snyder and Chris Kanich. No please, after you: Detecting fraud in affiliate marketing networks. In WEIS, 2015. URL: http://www.econinfosec.org/archive/weis2015/papers/WEIS_2015_snyder.pdf.
  60. K. Thompson. Programming techniques: Regular expression search algorithm. Communications of the ACM, 11, 1968. Google Scholar
  61. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5), 2018. URL: https://doi.org/10.1145/3186893.
  62. Georg Zetzsche. The complexity of downward closure comparisons. In ICALP, volume 55 of LIPIcs, 2016. URL: https://doi.org/10.4230/LIPICS.ICALP.2016.123.
  63. Haopeng Zhang, Yanlei Diao, and Neil Immerman. On complexity and optimization of expensive queries in complex event processing. In SIGMOD, 2014. URL: https://doi.org/10.1145/2588555.2593671.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact