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Extending the Burrows-Wheeler Transform for Cartesian Tree Matching and Constructing It

Authors Eric M. Osterkamp , Dominik Köppl

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  • Theory of computation
Keywords
  • Cartesian tree matching
  • extended Burrows-Wheeler transform
  • construction algorithm
  • generalized pattern matching

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Abstract

Cartesian tree matching is a form of generalized pattern matching where a substring of the text matches with the pattern if they share the same Cartesian tree. This form of matching finds application for time series of stock prices and can be of interest for melody matching between musical scores. For the indexing problem, the state-of-the-art data structure is a Burrows-Wheeler transform based solution due to [Kim and Cho, CPM'21], which uses nearly succinct space and can count the number of substrings that Cartesian tree match with a pattern in time linear in the pattern length. The authors address the construction of their data structure with a straight-forward solution that, however, requires pointer-based data structures, resulting in O(n lg n) bits of space, where n is the text length [Kim and Cho, CPM'21, Section A.4]. We address this bottleneck by a construction that requires O(n lg σ) bits of space and has a time complexity of O(n (lg σ lg n)/(lg lg n)), where σ is alphabet size. Additionally, we can extend this index for indexing multiple circular texts in the spirit of the extended Burrows-Wheeler transform without sacrificing the time and space complexities. We present this index in a dynamic variant, where we pay a logarithmic slowdown and need space linear in the input texts in bits for the extra functionality that we can incrementally add texts. Our extended setting is of interest for finding repetitive motifs common in the aforementioned applications, independent of offsets and scaling.

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Eric M. Osterkamp and Dominik Köppl. Extending the Burrows-Wheeler Transform for Cartesian Tree Matching and Constructing It. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CPM.2025.26

Author Details

Eric M. Osterkamp
  • University of Münster, Germany
Dominik Köppl
  • University of Yamanashi, Kofu, Japan

Funding

  • Köppl, Dominik: JSPS KAKENHI Grant Numbers JP23H04378 and JP25K21150.

Acknowledgements

We sincerely thank the anonymous reviewers for their constructive comments.

References

  1. Bastien Auvray, Julien David, Richard Groult, and Thierry Lecroq. Approximate cartesian tree matching: An approach using swaps. In Proc. SPIRE, volume 14240 of LNCS, pages 49-61, 2023. URL: https://doi.org/10.1007/978-3-031-43980-3_5.
  2. Christina Boucher, Davide Cenzato, Zsuzsanna Lipták, Massimiliano Rossi, and Marinella Sciortino. r-indexing the eBWT. In Proc. SPIRE, volume 12944 of LNCS, pages 3-12, 2021. URL: https://doi.org/10.1007/978-3-030-86692-1_1.
  3. Erik D. Demaine, Gad M. Landau, and Oren Weimann. On Cartesian trees and range minimum queries. Algorithmica, 68(3):610-625, 2014. URL: https://doi.org/10.1007/s00453-012-9683-x.
  4. Simone Faro, Thierry Lecroq, Kunsoo Park, and Stefano Scafiti. On the longest common Cartesian substring problem. The Computer Journal, 66(4):907-923, 2022. URL: https://doi.org/10.1093/COMJNL/BXAB204.
  5. Paolo Ferragina and Giovanni Manzini. Opportunistic data structures with applications. In Proc. FOCS, pages 390-398, 2000. URL: https://doi.org/10.1109/SFCS.2000.892127.
  6. Johannes Fischer. Optimal succinctness for range minimum queries. In Proc. LATIN, volume 6034 of LNCS, pages 158-169, 2010. URL: https://doi.org/10.1007/978-3-642-12200-2_16.
  7. Peter Foster, Anssi Klapuri, and Simon Dixon. A method for identifying repetition structure in musical audio based on time series prediction. In Proc. EUSIPCO, pages 1299-1303. IEEE, 2012. URL: https://ieeexplore.ieee.org/document/6334323/.
  8. Tak-Chung Fu, Korris Fu-Lai Chung, Robert Wing Pong Luk, and Chak-man Ng. Stock time series pattern matching: Template-based vs. rule-based approaches. Eng. Appl. Artif. Intell., 20(3):347-364, 2007. URL: https://doi.org/10.1016/J.ENGAPPAI.2006.07.003.
  9. Mitsuru Funakoshi, Takuya Mieno, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Computing maximal palindromes in non-standard matching models. In Proc. IWOCA, volume 14764 of LNCS, pages 165-179, 2024. URL: https://doi.org/10.1007/978-3-031-63021-7_13.
  10. Paweł Gawrychowski, Samah Ghazawi, and Gad M. Landau. On indeterminate strings matching. In Proc. CPM, volume 161 of LIPIcs, pages 14:1-14:14, 2020. URL: https://doi.org/10.4230/LIPICS.CPM.2020.14.
  11. Daiki Hashimoto, Diptarama Hendrian, Dominik Köppl, Ryo Yoshinaka, and Ayumi Shinohara. Computing the parameterized Burrows-Wheeler transform online. In Proc. SPIRE, volume 13617 of LNCS, pages 70-85, 2022. URL: https://doi.org/10.1007/978-3-031-20643-6_6.
  12. Yuzuru Hiraga. Structural recognition of music by pattern matching. In Proc. ICMC. Michigan Publishing, 1997. URL: https://hdl.handle.net/2027/spo.bbp2372.1997.113.
  13. Wing-Kai Hon, Tsung-Han Ku, Chen-Hua Lu, Rahul Shah, and Sharma V. Thankachan. Efficient algorithm for circular Burrows-Wheeler transform. In Proc. CPM, volume 7354 of LNCS, pages 257-268, 2012. URL: https://doi.org/10.1007/978-3-642-31265-6_21.
  14. Kento Iseri, Tomohiro I, Diptarama Hendrian, Dominik Köppl, Ryo Yoshinaka, and Ayumi Shinohara. Breaking a barrier in constructing compact indexes for parameterized pattern matching. In Proc. ICALP, volume 297 of LIPIcs, pages 89:1-89:19, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.89.
  15. Natsumi Kikuchi, Diptarama Hendrian, Ryo Yoshinaka, and Ayumi Shinohara. Computing covers under substring consistent equivalence relations. In Proc. SPIRE, volume 12303 of LNCS, pages 131-146, 2020. URL: https://doi.org/10.1007/978-3-030-59212-7_10.
  16. 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.
  17. Sung-Hwan Kim and Hwan-Gue Cho. A compact index for Cartesian tree matching. In Proc. CPM, volume 191 of LIPIcs, pages 18:1-18:19, 2021. URL: https://doi.org/10.4230/LIPIcs.CPM.2021.18.
  18. Sungmin Kim and Yo-Sub Han. Approximate Cartesian tree pattern matching. In Proc. DLT, volume 14791 of LNCS, pages 189-202, 2024. URL: https://doi.org/10.1007/978-3-031-66159-4_14.
  19. Marcin Kubica, Tomasz Kulczyński, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. A linear time algorithm for consecutive permutation pattern matching. Inf. Process. Lett., 113(12):430-433, 2013. URL: https://doi.org/10.1016/J.IPL.2013.03.015.
  20. Sabrina Mantaci, Antonio Restivo, Giovanna Rosone, and Marinella Sciortino. An extension of the Burrows-Wheeler transform. Theor. Comput. Sci., 387(3):298-312, 2007. URL: https://doi.org/10.1016/j.tcs.2007.07.014.
  21. 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.
  22. Gonzalo Navarro and Kunihiko Sadakane. Fully functional static and dynamic succinct trees. ACM Trans. Algorithms, 10(3):16:1-16:39, 2014. URL: https://doi.org/10.1145/2601073.
  23. Akio Nishimoto, Noriki Fujisato, Yuto Nakashima, and Shunsuke Inenaga. Position heaps for Cartesian-tree matching on strings and tries. In Proc. SPIRE, volume 12944 of LNCS, pages 241-254, 2021. URL: https://doi.org/10.1007/978-3-030-86692-1_20.
  24. Tsubasa Oizumi, Takeshi Kai, Takuya Mieno, Shunsuke Inenaga, and Hiroki Arimura. Cartesian tree subsequence matching. In Proc. CPM, volume 223 of LIPIcs, pages 14:1-14:18, 2022. URL: https://doi.org/10.4230/LIPICS.CPM.2022.14.
  25. Eric M. Osterkamp and Dominik Köppl. Extending the parameterized Burrows-Wheeler transform. In Proc. DCC, pages 143-152, 2024. URL: https://doi.org/10.1109/DCC58796.2024.00022.
  26. Sung Gwan Park, Magsarjav Bataa, Amihood Amir, Gad M. Landau, and Kunsoo Park. Finding patterns and periods in Cartesian tree matching. Theor. Comput. Sci., 845:181-197, 2020. URL: https://doi.org/10.1016/J.TCS.2020.09.014.
  27. 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.
  28. Jean Vuillemin. A unifying look at data structures. Commun. ACM, 23(4):229-239, 1980. URL: https://doi.org/10.1145/358841.358852.
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