Document Open Access Logo

Longest Lyndon Substring After Edit

Authors Yuki Urabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai , Masayuki Takeda

PDF

File

Thumbnail PDF
PDF
LIPIcs.CPM.2018.19.pdf
  • Filesize: 0.5 MB
  • 10 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Lyndon word
  • Lyndon factorization
  • Lyndon tree
  • Edit operation

Metrics

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

Abstract

The longest Lyndon substring of a string T is the longest substring of T which is a Lyndon word. LLS(T) denotes the length of the longest Lyndon substring of a string T. In this paper, we consider computing LLS(T') where T' is an edited string formed from T. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(log n) time for any single character edit. We also consider a version of the problem with block edits, i.e., a substring of T is replaced by a given string of length l. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(l log sigma + log n) time for any block edit where sigma is the number of distinct characters in T. We can modify our algorithm so as to output all the longest Lyndon substrings of T' for both problems.

Cite As Get BibTex

Yuki Urabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Longest Lyndon Substring After Edit. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 19:1-19:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CPM.2018.19

Author Details

Yuki Urabe
  • Department of Electrical Engineering and Computer Science, Kyushu University, Japan
Yuto Nakashima
  • Department of Informatics, Kyushu University, Japan
Shunsuke Inenaga
  • Department of Informatics, Kyushu University, Japan
Hideo Bannai
  • Department of Informatics, Kyushu University, Japan
Masayuki Takeda
  • Department of Informatics, Kyushu University, Japan

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP17H06923 (YN), JP17H01697 (SI), JP16H02783 (HB), and JP25240003 (MT).

References

  1. Amihood Amir, Panagiotis Charalampopoulos, Costas S. Iliopoulos, Solon P. Pissis, and Jakub Radoszewski. Longest common factor after one edit operation. In String Processing and Information Retrieval - 24th International Symposium, SPIRE 2017, Palermo, Italy, September 26-29, 2017, Proceedings, pages 14-26, 2017. Google Scholar
  2. Alberto Apostolico and Maxime Crochemore. Fast parallel Lyndon factorization with applications. Mathematical Systems Theory, 28(2):89-108, 1995. Google Scholar
  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. Google Scholar
  4. Frédérique Bassino, Julien Clément, and Cyril Nicaud. The standard factorization of Lyndon words: an average point of view. Discrete Mathematics, 290(1):1-25, 2005. Google Scholar
  5. Michael A. Bender and Martín Farach-Colton. The level ancestor problem simplified. TCS, 321(1):5-12, 2004. Google Scholar
  6. Dany Breslauer, Roberto Grossi, and Filippo Mignosi. Simple real-time constant-space string matching. In Proc. CPM 2011, pages 173-183, 2011. Google Scholar
  7. Marc Chemillier. Periodic musical sequences and Lyndon words. Soft Comput., 8(9):611-616, 2004. Google Scholar
  8. K. T. Chen, R. H. Fox, and R. C. Lyndon. Free differential calculus. iv. the quotient groups of the lower central series. Annals of Mathematics, 68(1):81-95, 1958. Google Scholar
  9. Maxime Crochemore and Dominique Perrin. Two-way string matching. J. ACM, 38(3):651-675, 1991. Google Scholar
  10. Jacqueline W. Daykin, Frantisek Franek, Jan Holub, A. S. M. Sohidull Islam, and W. F. Smyth. Reconstructing a string from its Lyndon arrays. Theor. Comput. Sci., 710:44-51, 2018. Google Scholar
  11. Jacqueline W. Daykin, Costas S. Iliopoulos, and William F. Smyth. Parallel RAM algorithms for factorizing words. Theor. Comput. Sci., 127(1):53-67, 1994. Google Scholar
  12. Olivier Delgrange and Eric Rivals. STAR: an algorithm to search for tandem approximate repeats. Bioinformatics, 20(16):2812-2820, 2004. Google Scholar
  13. Jean-Pierre Duval. Factorizing words over an ordered alphabet. J. Algorithms, 4(4):363-381, 1983. Google Scholar
  14. Frantisek Franek, A. S. M. Sohidull Islam, Mohammad Sohel Rahman, and William F. Smyth. Algorithms to compute the Lyndon array. In Proceedings of the Prague Stringology Conference 2016, Prague, Czech Republic, August 29-31, 2016, pages 172-184, 2016. Google Scholar
  15. Harold Fredricksen and James Maiorana. Necklaces of beads in k colors and k-ary de Bruijn sequences. Discrete Mathematics, 23(3):207-210, 1978. Google Scholar
  16. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984. Google Scholar
  17. Christophe Hohlweg and Christophe Reutenauer. Lyndon words, permutations and trees. Theor. Comput. Sci., 307(1):173-178, 2003. URL: http://dx.doi.org/10.1016/S0304-3975(03)00099-9.
  18. Tomohiro I, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Faster Lyndon factorization algorithms for SLP and LZ78 compressed text. Theor. Comput. Sci., 656:215-224, 2016. Google Scholar
  19. M. Lothaire. Combinatorics on Words. Addison-Wesley, 1983. Google Scholar
  20. R. C. Lyndon. On Burnside’s problem. Transactions of the American Mathematical Society, 77:202-215, 1954. Google Scholar
  21. Marcin Mucha. Lyndon words and short superstrings. In Proc. SODA'13, pages 958-972, 2013. Google Scholar
  22. Yuto Nakashima, Takuya Takagi, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. On reverse engineering the Lyndon tree. In Proceedings of the Prague Stringology Conference 2017, Prague, Czech Republic, August 28-30, 2017, pages 108-117, 2017. Google Scholar
  23. Shoshana Neuburger and Dina Sokol. Succinct 2D dictionary matching. Algorithmica, pages 1-23, 2012. 10.1007/s00453-012-9615-9. Google Scholar
  24. Xavier Provençal. Minimal non-convex words. Theor. Comput. Sci., 412(27):3002-3009, 2011. Google Scholar
  25. E. Ukkonen. On-line construction of suffix trees. Algorithmica, 14(3):249-260, 1995. Google Scholar
  26. P. Weiner. Linear pattern-matching algorithms. In Proc. of 14th IEEE Ann. Symp. on Switching and Automata Theory, pages 1-11. Institute of Electrical Electronics Engineers, New York, 1973. Google Scholar
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