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Palindromic k-Factorization in Pure Linear Time

Authors Mikhail Rubinchik, Arseny M. Shur

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Author Details

Mikhail Rubinchik
  • Ural Federal University, Ekaterinburg, Russia
Arseny M. Shur
  • Ural Federal University, Ekaterinburg, Russia

Funding

Supported by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1).

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 81:1-81:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.81

Abstract

Given a string s of length n over a general alphabet and an integer k, the problem is to decide whether s is a concatenation of k nonempty palindromes. Two previously known solutions for this problem work in time O(kn) and O(nlog n) respectively. Here we settle the complexity of this problem in the word-RAM model, presenting an O(n)-time online deciding algorithm. The algorithm simultaneously finds the minimum odd number of factors and the minimum even number of factors in a factorization of a string into nonempty palindromes. We also demonstrate how to get an explicit factorization of s into k palindromes with an O(n)-time offline postprocessing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • stringology
  • palindrome
  • palindromic factorization
  • online algorithm

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References

  1. V. Arlazarov, E. Dinic, M. Kronrod, and I. Faradzev. On economical construction of the transitive closure of a directed graph. Dokl. Akad. Nauk SSSR, 194(11):1209-1210, 1970. Google Scholar
  2. Hideo Bannai, Travis Gagie, Shunsuke Inenaga, Juha Kärkkäinen, Dominik Kempa, Marcin Piatkowski, Simon J. Puglisi, and Shiho Sugimoto. Diverse palindromic factorization is NP-complete. In Developments in Language Theory - 19th International Conference, DLT 2015, Liverpool, UK, July 27-30, 2015, Proceedings, volume 9168 of Lecture Notes in Computer Science, pages 85-96. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21500-6_6.
  3. Kirill Borozdin, Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Palindromic length in linear time. In 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, July 4-6, 2017, Warsaw, Poland, volume 78 of LIPIcs, pages 23:1-23:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CPM.2017.23.
  4. G. Fici, T. Gagie, J. Kärkkäinen, and D. Kempa. A subquadratic algorithm for minimum palindromic factorization. J. of Discrete Algorithms, 28:41-48, 2014. URL: https://doi.org/10.1016/j.jda.2014.08.001.
  5. N. J. Fine and H. S. Wilf. Uniqueness theorems for periodic functions. Proceedings of the American Mathematical Society, 16(1):109-114, 1965. Google Scholar
  6. Anna E. Frid. Sturmian numeration systems and decompositions to palindromes. Eur. J. Comb., 71:202-212, 2018. URL: https://doi.org/10.1016/j.ejc.2018.04.003.
  7. Anna E. Frid, Svetlana Puzynina, and Luca Q. Zamboni. On palindromic factorization of words. Adv. Appl. Math., 50(5):737-748, 2013. URL: https://doi.org/10.1016/j.aam.2013.01.002.
  8. Z. Galil and J. Seiferas. A linear-time on-line recognition algorithm for "Palstar". J. ACM, 25:102-111, 1978. Google Scholar
  9. T. I, S. Sugimoto, S. Inenaga, H. Bannai, and M. Takeda. Computing palindromic factorizations and palindromic covers on-line. In CPM 2014, volume 8486 of LNCS, pages 150-161. Springer, 2014. Google Scholar
  10. R. C. Lyndon K.-T. Chen, R. H. Fox. Free differential calculus. iv. the quotient groups of the lower central series. Annals of Mathematics, 2nd Ser., 68(1):81-95, 1958. URL: https://doi.org/10.2307/1970044.
  11. D. E. Knuth, J. Morris, and V. Pratt. Fast pattern matching in strings. SIAM J. Comp., 6:323-350, 1977. Google Scholar
  12. D. Kosolobov, M. Rubinchik, and A. M. Shur. Pal^k is linear recognizable online. In SOFSEM 2015, volume 8939 of LNCS, pages 289-301. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-46078-8_24.
  13. A. Lempel and J. Ziv. On the complexity of finite sequences. IEEE Transactions on Information Theory, 22(1):75-81, 1976. Google Scholar
  14. G. Manacher. A new linear-time on-line algorithm finding the smallest initial palindrome of a string. J. of the ACM, 22(3):346-351, 1975. Google Scholar
  15. Yoshiaki Matsuoka, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda, and Florin Manea. Factorizing a string into squares in linear time. In 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, volume 54 of LIPIcs, pages 27:1-27:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  16. 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.
  17. Aleksi Saarela. Palindromic length in free monoids and free groups. In Combinatorics on Words - 11th International Conference, WORDS 2017, Montréal, QC, Canada, September 11-15, 2017, Proceedings, volume 10432 of Lecture Notes in Computer Science, pages 203-213. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-66396-8_19.
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