Document Open Access Logo

Compact Representation of Maximal Palindromes

Author Takuya Mieno

PDF

File

Thumbnail PDF
PDF
LIPIcs.CPM.2026.4.pdf
  • Filesize: 0.89 MB
  • 12 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • palindromes
  • succinct data structures
  • internal queries

Metrics

Abstract

Palindromes are strings that read the same forward and backward. The computation of palindromic structures within strings is a fundamental problem in string algorithms, being motivated by potential applications in formal language theory and bioinformatics. Although the number of palindromic factors in a string of length n can be quadratic, they can be implicitly represented in O(n log n) bits of space by storing the lengths of all maximal palindromes in an integer array, which can be computed in O(n) time [Manacher, 1975]. In this paper, we propose a novel O(n)-bit representation of all maximal palindromes in a string, which enables O(1)-time retrieval of the length of the maximal palindrome centered at any given position. The data structure can be constructed in O(n) time from the input string of length n. Since Manacher’s algorithm and the notion of maximal palindromes are widely utilized for solving numerous problems involving palindromic structures, our compact representation will accelerate the development of more space-efficient solutions to such problems. Indeed, as the first application of our compact representation of maximal palindromes, we present a data structure of size O(n) bits that can compute the longest palindrome appearing in any given factor of a string of length n in O(log n) time.

Cite As Get BibTex

Takuya Mieno. Compact Representation of Maximal Palindromes. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.4

Author Details

Takuya Mieno
  • Department of Computer and Network Engineering, University of Electro-Communications, Chofu, Japan

Funding

JSPS KAKENHI Grant Number JP24K20734

References

  1. Amihood Amir, Panagiotis Charalampopoulos, Solon P. Pissis, and Jakub Radoszewski. Dynamic and internal longest common substring. Algorithmica, 82(12):3707-3743, 2020. URL: https://doi.org/10.1007/S00453-020-00744-0.
  2. Alberto Apostolico, Dany Breslauer, and Zvi Galil. Parallel detection of all palindromes in a string. Theor. Comput. Sci., 141(1&2):163-173, 1995. URL: https://doi.org/10.1016/0304-3975(94)00083-U.
  3. Panagiotis Charalampopoulos, Solon P. Pissis, and Jakub Radoszewski. Longest palindromic substring in sublinear time. In 33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022, June 27-29, 2022, Prague, Czech Republic, volume 223 of LIPIcs, pages 20:1-20:9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CPM.2022.20.
  4. Xavier Droubay, Jacques Justin, and Giuseppe Pirillo. Episturmian words and some constructions of de Luca and Rauzy. Theor. Comput. Sci., 255(1-2):539-553, 2001. URL: https://doi.org/10.1016/S0304-3975(99)00320-5.
  5. Jonas Ellert, Pawel Gawrychowski, and Tatiana Starikovskaya. Minimal generators in optimal time. In 36th Annual Symposium on Combinatorial Pattern Matching, CPM 2025, June 17-19, 2025, Milan, Italy, volume 331 of LIPIcs, pages 14:1-14:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.CPM.2025.14.
  6. Nathan J Fine and Herbert S Wilf. Uniqueness theorems for periodic functions. Proceedings of the American Mathematical Society, 16(1):109-114, 1965. URL: https://doi.org/10.1090/S0002-9939-1965-0174934-9.
  7. Johannes Fischer and Volker Heun. Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput., 40(2):465-492, 2011. URL: https://doi.org/10.1137/090779759.
  8. Mitsuru Funakoshi, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Computing longest palindromic substring after single-character or block-wise edits. Theor. Comput. Sci., 859:116-133, 2021. URL: https://doi.org/10.1016/J.TCS.2021.01.014.
  9. Zvi Galil and Joel I. Seiferas. Recognizing certain repetitions and reversals within strings. In 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, USA, 25-27 October 1976, pages 236-252. IEEE Computer Society, 1976. URL: https://doi.org/10.1109/SFCS.1976.25.
  10. Dan Gusfield. Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology. Cambridge University Press, 1997. URL: https://doi.org/10.1017/CBO9780511574931.
  11. 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.
  12. Michael Itzhaki. Palindromes compression and retrieval. In Data Compression Conference, DCC 2025, Snowbird, UT, USA, March 18-21, 2025, page 375. IEEE, 2025. URL: https://doi.org/10.1109/DCC62719.2025.00062.
  13. 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.
  14. Roman Kolpakov and Gregory Kucherov. Searching for gapped palindromes. Theor. Comput. Sci., 410(51):5365-5373, 2009. URL: https://doi.org/10.1016/J.TCS.2009.09.013.
  15. Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Pal^k is linear recognizable online. In SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science, Pec pod Sněžkou, Czech Republic, January 24-29, 2015. Proceedings, volume 8939 of Lecture Notes in Computer Science, pages 289-301. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-46078-8_24.
  16. Stefan Langerman and Andrew Winslow. A quasilinear-time algorithm for tiling the plane isohedrally with a polyomino. In 32nd International Symposium on Computational Geometry, SoCG 2016, June 14-18, 2016, Boston, MA, USA, volume 51 of LIPIcs, pages 50:1-50:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.SOCG.2016.50.
  17. Veli Mäkinen, Djamal Belazzougui, Fabio Cunial, and Alexandru I. Tomescu. Genome-Scale Algorithm Design: Bioinformatics in the Era of High-Throughput Sequencing (2nd edition). Cambridge University Press, 2023. URL: http://www.genome-scale.info/.
  18. Glenn K. Manacher. A new linear-time "on-line" algorithm for finding the smallest initial palindrome of a string. J. ACM, 22(3):346-351, 1975. URL: https://doi.org/10.1145/321892.321896.
  19. Wataru Matsubara, Shunsuke Inenaga, Akira Ishino, Ayumi Shinohara, Tomoyuki Nakamura, and Kazuo Hashimoto. Efficient algorithms to compute compressed longest common substrings and compressed palindromes. Theor. Comput. Sci., 410(8-10):900-913, 2009. URL: https://doi.org/10.1016/J.TCS.2008.12.016.
  20. Takuya Mieno and Mitsuru Funakoshi. Data structures for computing unique palindromes in static and non-static strings. Algorithmica, 86(3):852-873, 2024. URL: https://doi.org/10.1007/S00453-023-01170-8.
  21. Takuya Mieno, Mitsuru Funakoshi, and Shunsuke Inenaga. Computing palindromes on a trie in linear time. In 33rd International Symposium on Algorithms and Computation, ISAAC 2022, December 19-21, 2022, Seoul, Korea, volume 248 of LIPIcs, pages 15:1-15:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2022.15.
  22. Kazuki Mitani, Takuya Mieno, Kazuhisa Seto, and Takashi Horiyama. Finding top-k longest palindromes in substrings. Theor. Comput. Sci., 979:114183, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114183.
  23. Shintaro Narisada, Diptarama Hendrian, Ryo Yoshinaka, and Ayumi Shinohara. Linear-time online algorithm for inferring the shortest path graph from a walk label. Theor. Comput. Sci., 812:187-202, 2020. URL: https://doi.org/10.1016/J.TCS.2019.10.029.
  24. Mikhail Rubinchik and Arseny M. Shur. Counting palindromes in substrings. In String Processing and Information Retrieval - 24th International Symposium, SPIRE 2017, Palermo, Italy, September 26-29, 2017, Proceedings, volume 10508 of Lecture Notes in Computer Science, pages 290-303. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-67428-5_25.
  25. Mikhail Rubinchik and Arseny M. Shur. EERTREE: an efficient data structure for processing palindromes in strings. Eur. J. Comb., 68:249-265, 2018. URL: https://doi.org/10.1016/J.EJC.2017.07.021.
  26. Mikhail Rubinchik and Arseny M. Shur. Palindromic k-factorization in pure linear time. In 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 81:1-81:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.81.
  27. Michael Sipser. Introduction to the theory of computation. PWS Publishing Company, 1997. Google Scholar
  28. Peter Weiner. Linear pattern matching algorithms. In 14th Annual Symposium on Switching and Automata Theory, Iowa City, Iowa, USA, October 15-17, 1973, pages 1-11. IEEE Computer Society, 1973. URL: https://doi.org/10.1109/SWAT.1973.13.
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