Document Open Access Logo

Conditional Lower Bounds for String Matching in Labelled Graphs

Author Massimo Equi

PDF

File

Thumbnail PDF
PDF
OASIcs.Grossi.7.pdf
  • Filesize: 0.97 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Theory of computation → Problems, reductions and completeness
Keywords
  • conditional lower bounds
  • strong exponential time hypothesis
  • fine-grained complexity
  • string matching
  • graphs

Metrics

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

Abstract

The problem of String Matching in Labelled Graphs (SMLG) is one possible generalization of the classic problem of finding a string inside another of greater length. In its most general form, SMLG asks to find a match for a string into a graph, which can be directed or undirected. As for string matching, many different variations are possible. For example, the match could be exact or approximate, and the match could lie on a path or a walk. Some of these variations easily fall into the NP-hard realm, while other variants are solvable in polynomial time. For the latter ones, fine-grained complexity has been a game changer in proving quadratic conditional lower bounds, allowing to finally close the gap with those upper bounds that remained unmatched for almost two decades.
If the match is allowed to be approximate, SMLG enjoys the same conditional quadratic lower bounds shown for example for edit distance (Backurs and Indyk, STOC '15). The case that really requires ad hoc conditional lower bounds is the one of finding an exact match that lies on a walk. In this work, we focus on explaining various conditional lower bounds for this version of SMLG, with the goal of giving an overall perspective that could help understand which aspects of the problem make it quadratic. We will introduce the reader to the field of fine-grained complexity and show how it can successfully provide the exact type of lower bounds needed for polynomial problems such as SMLG.

Cite As Get BibTex

Massimo Equi. Conditional Lower Bounds for String Matching in Labelled Graphs. In From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 132, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/OASIcs.Grossi.7

Author Details

Massimo Equi
  • Aalto University, Finland

Funding

This work was supported in part by the Research Council of Finland, Grant 359104.

References

  1. Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 375-388. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897653.
  2. Tatsuya Akutsu. A linear time pattern matching algorithm between a string and a tree. In Alberto Apostolico, Maxime Crochemore, Zvi Galil, and Udi Manber, editors, Combinatorial Pattern Matching, 4th Annual Symposium, CPM 93, Padova, Italy, June 2-4, 1993, Proceedings, volume 684 of Lecture Notes in Computer Science, pages 1-10. Springer, 1993. URL: https://doi.org/10.1007/BFb0029792.
  3. Amihood Amir, Moshe Lewenstein, and Noa Lewenstein. Pattern matching in hypertext. In Frank K. H. A. Dehne, Andrew Rau-Chaplin, Jörg-Rüdiger Sack, and Roberto Tamassia, editors, Algorithms and Data Structures, 5th International Workshop, WADS '97, Halifax, Nova Scotia, Canada, August 6-8, 1997, Proceedings, volume 1272 of Lecture Notes in Computer Science, pages 160-173. Springer, 1997. URL: https://doi.org/10.1007/3-540-63307-3_56.
  4. Amihood Amir, Moshe Lewenstein, and Noa Lewenstein. Pattern matching in hypertext. J. Algorithms, 35(1):82-99, 2000. URL: https://doi.org/10.1006/jagm.1999.1063.
  5. Renzo Angles and Claudio Gutierrez. Survey of graph database models. ACM Comput. Surv., 40(1):1:1-1:39, February 2008. URL: https://doi.org/10.1145/1322432.1322433.
  6. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). SIAM J. Comput., 47(3):1087-1097, 2018. URL: https://doi.org/10.1137/15M1053128.
  7. Manuel Cáceres. Parameterized algorithms for string matching to dags: Funnels and beyond. In Laurent Bulteau and Zsuzsanna Lipták, editors, 34th Annual Symposium on Combinatorial Pattern Matching, CPM 2023, June 26-28, 2023, Marne-la-Vallée, France, volume 259 of LIPIcs, pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.CPM.2023.7.
  8. Nicola Cotumaccio, Giovanna D'Agostino, Alberto Policriti, and Nicola Prezza. Co-lexicographically ordering automata and regular languages - part I. J. ACM, 70(4):27:1-27:73, 2023. URL: https://doi.org/10.1145/3607471.
  9. Nicola Cotumaccio and Nicola Prezza. On indexing and compressing finite automata. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2585-2599. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.153.
  10. Massimo Equi, Roberto Grossi, Veli Mäkinen, and Alexandru I. Tomescu. On the complexity of string matching for graphs. In 46th International Colloquium on Automata, Languages, and Programming (ICALP), volume 132 of LIPIcs, pages 55:1-55:15, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.55.
  11. Massimo Equi, Veli Mäkinen, and Alexandru I. Tomescu. Graphs cannot be indexed in polynomial time for sub-quadratic time string matching, unless SETH fails. Theor. Comput. Sci., 975:114128, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114128.
  12. Massimo Equi, Veli Mäkinen, Alexandru I. Tomescu, and Roberto Grossi. On the complexity of string matching for graphs. ACM Trans. Algorithms, 19(3):21:1-21:25, 2023. URL: https://doi.org/10.1145/3588334.
  13. Massimo Equi, Tuukka Norri, Jarno Alanko, Bastien Cazaux, Alexandru I. Tomescu, and Veli Mäkinen. Algorithms and complexity on indexing elastic founder graphs. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ISAAC.2021.20.
  14. Travis Gagie, Giovanni Manzini, and Jouni Sirén. Wheeler graphs: A framework for bwt-based data structures. Theor. Comput. Sci., 698:67-78, 2017. URL: https://doi.org/10.1016/J.TCS.2017.06.016.
  15. Garrison Erik, Sirén Jouni, Novak Adam M, Hickey Glenn, Eizenga Jordan M, Dawson Eric T, Jones William, Garg Shilpa, Markello Charles, Lin Michael F, Paten Benedict, and Durbin Richard. Variation graph toolkit improves read mapping by representing genetic variation in the reference. Nature Biotechnology, 36:875, August 2018. URL: https://doi.org/10.1038/nbt.4227 10.1038/nbt.4227.
  16. Daniel Gibney, Gary Hoppenworth, and Sharma V. Thankachan. Simple reductions from formula-sat to pattern matching on labeled graphs and subtree isomorphism. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 232-242. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.26.
  17. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  18. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  19. Donald E. Knuth, James H. Morris Jr., and Vaughan R. Pratt. Fast pattern matching in strings. SIAM J. Comput., 6(2):323-350, 1977. URL: https://doi.org/10.1137/0206024.
  20. Udi Manber and Sun Wu. Approximate string matching with arbitrary costs for text and hypertext. In Advances In Structural And Syntactic Pattern Recognition, Bern, Switzerland, 26-28 August 1992, pages 22-33. World Scientific, 1992. URL: https://doi.org/10.1142/9789812797919_0002.
  21. Gonzalo Navarro. Improved approximate pattern matching on hypertext. Theor. Comput. Sci., 237(1-2):455-463, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00333-3.
  22. Kunsoo Park and Dong Kyue Kim. String matching in hypertext. In Zvi Galil and Esko Ukkonen, editors, Combinatorial Pattern Matching, 6th Annual Symposium, CPM 95, Espoo, Finland, July 5-7, 1995, Proceedings, volume 937 of Lecture Notes in Computer Science, pages 318-329. Springer, 1995. URL: https://doi.org/10.1007/3-540-60044-2_51.
  23. Nicola Rizzo, Massimo Equi, Tuukka Norri, and Veli Mäkinen. Elastic founder graphs improved and enhanced. Theor. Comput. Sci., 982:114269, 2024. URL: https://doi.org/10.1016/J.TCS.2023.114269.
  24. Chuan Shi, Yitong Li, Jiawei Zhang, Yizhou Sun, and Philip S. Yu. A survey of heterogeneous information network analysis. IEEE Trans. Knowl. Data Eng., 29(1):17-37, 2017. URL: https://doi.org/10.1109/TKDE.2016.2598561.
  25. Mikko Rautiainen Tobias and Marschall. Aligning sequences to general graphs in O(V + mE) time. bioRxiv, 2017. URL: https://doi.org/10.1101/216127.
  26. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. URL: https://doi.org/10.1016/J.TCS.2005.09.023.
  27. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the International Congress of Mathematicians (ICM 2018), pages 3447-3487, 2018. URL: https://doi.org/10.1142/9789813272880_0188.
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