On the Maximal Number of Cubic Runs in a String

  • Maxime Crochemore
  • Costas Iliopoulos
  • Marcin Kubica
  • Jakub Radoszewski
  • Wojciech Rytter
  • Tomasz Waleń
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)


A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ |v|. The maximal number of runs in a string of length n has been thoroughly studied, and is known to be between 0.944 n and 1.029 n. In this paper we investigate cubic runs, in which the shortest period p satisfies 3p ≤ |v|. We show the upper bound of 0.5 n on the maximal number of such runs in a string of length n, and construct an infinite sequence of words over binary alphabet for which the lower bound is 0.406 n.


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  1. 1.
    Baturo, P., Piatkowski, M., Rytter, W.: The number of runs in Sturmian words. In: Ibarra, O.H., Ravikumar, B. (eds.) CIAA 2008. LNCS, vol. 5148, pp. 252–261. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Berstel, J., Karhumäki, J.: Combinatorics on words: a tutorial. Bulletin of the EATCS 79, 178–228 (2003)zbMATHGoogle Scholar
  3. 3.
    Crochemore, M., Ilie, L.: Analysis of maximal repetitions in strings. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 465–476. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Crochemore, M., Ilie, L.: Maximal repetitions in strings. J. Comput. Syst. Sci. 74(5), 796–807 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Crochemore, M., Ilie, L., Rytter, W.: Repetitions in strings: Algorithms and combinatorics. Theor. Comput. Sci. 410(50), 5227–5235 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Crochemore, M., Ilie, L., Tinta, L.: Towards a solution to the “runs” conjecture. In: Ferragina, P., Landau, G.M. (eds.) CPM 2008. LNCS, vol. 5029, pp. 290–302. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Crochemore, M., Rytter, W.: Squares, cubes, and time-space efficient string searching. Algorithmica 13(5), 405–425 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Franek, F., Yang, Q.: An asymptotic lower bound for the maximal number of runs in a string. Int. J. Found. Comput. Sci. 19(1), 195–203 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Giraud, M.: Not so many runs in strings. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 232–239. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Kolpakov, R.M., Kucherov, G.: Finding maximal repetitions in a word in linear time. In: Proceedings of the 40th Symposium on Foundations of Computer Science, pp. 596–604 (1999)Google Scholar
  11. 11.
    Kubica, M., Radoszewski, J., Rytter, W., Walen, T.: On the maximal number of cubic subwords in a string. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 345–355. Springer, Heidelberg (2009)Google Scholar
  12. 12.
    Kusano, K., Matsubara, W., Ishino, A., Bannai, H., Shinohara, A.: New lower bounds for the maximum number of runs in a string. CoRR abs/0804.1214 (2008)Google Scholar
  13. 13.
    Lothaire, M.: Combinatorics on Words. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  14. 14.
    Mignosi, F., Pirillo, G.: Repetitions in the Fibonacci infinite word. ITA 26, 199–204 (1992)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Rytter, W.: The number of runs in a string: Improved analysis of the linear upper bound. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 184–195. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Rytter, W.: The structure of subword graphs and suffix trees in Fibonacci words. Theor. Comput. Sci. 363(2), 211–223 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rytter, W.: The number of runs in a string. Inf. Comput. 205(9), 1459–1469 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Simpson, J.: Modified Padovan words and the maximum number of runs in a word. Australasian Journal of Combinatorics (to appear, 2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Maxime Crochemore
    • 1
    • 3
  • Costas Iliopoulos
    • 1
    • 4
  • Marcin Kubica
    • 2
  • Jakub Radoszewski
    • 2
  • Wojciech Rytter
    • 2
    • 5
  • Tomasz Waleń
    • 2
  1. 1.King’s College LondonLondonUK
  2. 2.Dept. of Mathematics, Computer Science and MechanicsUniversity of WarsawWarsawPoland
  3. 3.Université Paris-EstFrance
  4. 4.Digital Ecosystems & Business Intelligence InstituteCurtin University of TechnologyPerthAustralia
  5. 5.Dept. of Math. and InformaticsCopernicus UniversityToruńPoland

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