Document Open Access Logo

Efficient Index for Square Pattern Matching

Authors Po-Chun Chen , Che-Wei Tsao , Wing-Kai Hon , Dominik Köppl

PDF

File

Thumbnail PDF
PDF
LIPIcs.CPM.2026.35.pdf
  • Filesize: 1.06 MB
  • 12 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • string algorithms
  • pattern matching
  • indexing
  • squares

Metrics

Abstract

A string S is called a square if it can be written as the concatenation of two identical strings. Two strings P and Q of the same length are said to square match if, for every substring of P, it is a square if and only if the corresponding substring of Q is also a square. The square pattern matching problem asks for locating all substrings of a given text T of length n that square match a query pattern P of length m. This notion captures similarity in repetition structures and is motivated by applications in areas such as bioinformatics and music structure analysis.
In this paper, we introduce a novel technique, called the longest prefix square (LPS) encoding, which represents the square structure of a string as an integer array of the same length. We show that two strings square match if and only if they have identical LPS encodings. Based on this result, we construct an index solving the square pattern matching problem in time O(m lg m + occ) using O(nlg²n) bits of space, where occ denotes the number of occurrences of substrings in T that square match P. If the LPS encoding of P is precomputed, the query time improves to O(m + occ).

Cite As Get BibTex

Po-Chun Chen, Che-Wei Tsao, Wing-Kai Hon, and Dominik Köppl. Efficient Index for Square Pattern Matching. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 35:1-35:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.35

Author Details

Po-Chun Chen
  • National Tsing Hua University, Hsinchu, Taiwan
Che-Wei Tsao
  • National Tsing Hua University, Hsinchu, Taiwan
Wing-Kai Hon
  • National Tsing Hua University, Hsinchu, Taiwan
Dominik Köppl
  • University of Yamanashi, Kofu, Japan

Funding

  • Hon, Wing-Kai: This work was supported by Taiwan NSTC Grant Number 113-2221-E-007-139.
  • Köppl, Dominik: This work was supported by JSPS KAKENHI Grant Number 25K21150.

Acknowledgements

We thank the anonymous reviewers for their comments, which greatly strengthen our results.

References

  1. Amihood Amir, Richard Cole, Ramesh Hariharan, Moshe Lewenstein, and Ely Porat. Overlap matching. Inf. Comput., 181(1):57-74, 2003. URL: https://doi.org/10.1016/S0890-5401(02)00035-4.
  2. Brenda S. Baker. Parameterized Pattern Matching: Algorithms and Applications. J. Comput. Syst. Sci., 52(1):28-42, 1996. URL: https://doi.org/10.1006/JCSS.1996.0003.
  3. Hideo Bannai, Tomohiro I, Shunsuke Inenaga, Yuto Nakashima, Masayuki Takeda, and Kazuya Tsuruta. The "Runs" Theorem. SIAM J. Comput., 46(5):1501-1514, 2017. URL: https://doi.org/10.1137/15M1011032.
  4. Maxime Crochemore and Wojciech Rytter. Squares, Cubes, and Time-Space Efficient String Searching. Algorithmica, 13(5):405-425, 1995. URL: https://doi.org/10.1007/BF01190846.
  5. Maxime Crochemore and Wojciech Rytter. Jewels of stringology: text algorithms. World Scientific, 2002. Google Scholar
  6. Jonas Ellert and Johannes Fischer. Linear Time Runs Over General Ordered Alphabets. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, Glasgow, Scotland (Virtual Conference), July 12-16, 2021, pages 63:1-63:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.63.
  7. Michael Fredman and Dan Willard. Blasting through the information theoretic barrier with fusion trees. In ACM Symposium on Theory of Computing, pages 1-7, 1990. URL: https://doi.org/10.1145/100216.100217.
  8. Garance Gourdel, Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, Arseny M. Shur, and Tomasz Walen. String periods in the order-preserving model. Inf. Comput., 270, 2020. URL: https://doi.org/10.1016/J.IC.2019.104463.
  9. Diptarama Hendrian. Generalized Dictionary Matching Under Substring Consistent Equivalence Relations. In WALCOM: Algorithms and Computation (WALCOM 2020), volume 12049 of Lecture Notes in Computer Science, pages 120-132. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-39881-1_11.
  10. Tomohiro I, Shunsuke Inenaga, and Masayuki Takeda. Palindrome pattern matching. Theor. Comput. Sci., 483:162-170, 2013. URL: https://doi.org/10.1016/J.TCS.2012.01.047.
  11. Jinil Kim, Peter Eades, Rudolf Fleischer, Seok-Hee Hong, Costas S. Iliopoulos, Kunsoo Park, Simon J. Puglisi, and Takeshi Tokuyama. Order-preserving matching. Theor. Comput. Sci., 525:68-79, 2014. URL: https://doi.org/10.1016/J.TCS.2013.10.006.
  12. 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.
  13. Roman M. Kolpakov and Gregory Kucherov. Finding Maximal Repetitions in a Word in Linear Time. In 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, New York, NY, USA, October 17-18, 1999, pages 596-604. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814634.
  14. Xingyu Liao, Wufei Zhu, Juexiao Zhou, Haoyang Li, Xiaopeng Xu, Bin Zhang, and Xin Gao. Repetitive DNA sequence detection and its role in the human genome. Communications biology, 6(1):954, 2023. URL: https://doi.org/10.1038/s42003-023-05322-y.
  15. Yoshiaki Matsuoka, Takahiro Aoki, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Generalized pattern matching and periodicity under substring consistent equivalence relations. Theor. Comput. Sci., 656:225-233, 2016. URL: https://doi.org/10.1016/J.TCS.2016.02.017.
  16. Peter Bro Miltersen. Error Correcting Codes, Perfect Hashing Circuits, and Deterministic Dynamic Dictionaries. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 556-563. SIAM, 1998. URL: http://dl.acm.org/citation.cfm?id=314613.314845.
  17. Shinya Nagashita and Tomohiro I. PalFM-Index: FM-Index for Palindrome Pattern Matching. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023), volume 259 of LIPIcs, pages 23:1-23:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CPM.2023.23.
  18. Sung Gwan Park, Amihood Amir, Gad M. Landau, and Kunsoo Park. Cartesian Tree Matching and Indexing. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019), volume 128 of LIPIcs, pages 16:1-16:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CPM.2019.16.
  19. Mihai Pătraşcu and Mikkel Thorup. Dynamic Integer Sets with Optimal Rank, Select, and Predecessor Search. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 166-175. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.26.
  20. Jouni Paulus and Anssi Klapuri. Music structure analysis by finding repeated parts. In Proceedings of the 1st ACM Workshop on Audio and Music Computing Multimedia, pages 59-68. Association for Computing Machinery, 2006. URL: https://doi.org/10.1145/1178723.1178733.
  21. Jouni Paulus, Meinard Müller, and Anssi Klapuri. State of the Art Report: Audio-Based Music Structure Analysis. In International Society for Music Information Retrieval Conference (ISMIR 2010), pages 625-636. ISMIR, 2010. URL: http://ismir2010.ismir.net/proceedings/ismir2010-107.pdf.
  22. Siwoo Song, Geonmo Gu, Cheol Ryu, Simone Faro, Thierry Lecroq, and Kunsoo Park. Fast algorithms for single and multiple pattern Cartesian tree matching. Theor. Comput. Sci., 849:47-63, 2021. URL: https://doi.org/10.1016/J.TCS.2020.10.009.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact