Bounds on Powers in Strings

  • Maxime Crochemore
  • Szilárd Zsolt Fazekas
  • Costas Iliopoulos
  • Inuka Jayasekera
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5257)

Abstract

We show a Θ(n logn) bound on the maximal number of occurrences of primitively-rooted k-th powers occurring in a string of length n for any integer k, k ≥ 2. We also show a Θ(n 2) bound on the maximal number of primitively-rooted powers with fractional exponent e, 1 < e < 2, occurring in a string of length n. This result holds obviously for their maximal number of occurrences. The first result contrasts with the linear number of occurrences of maximal repetitions of exponent at least 2.

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References

  1. 1.
    Apostolico, A., Preparata, F.P.: Optimal off-line detection of repetitions in a string. Theoret. Comput. Sci. 22(3), 297–315 (1983)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Crochemore, M.: An optimal algorithm for computing the repetitions in a word. Inf. Process. Lett. 12(5), 244–250 (1981)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Crochemore, M., Ilie, L.: Maximal repetitions in strings. J. Comput. Syst. Sci. (in press, 2007)Google Scholar
  4. 4.
    Crochemore, M., Ilie, L., Tinta, L.: Towards a solution to the “runs” conjecture. In: Ferragina, P., Landau, G.M. (eds.) Combinatorial Pattern Matching. LNCS. Springer, Berlin (in press, 2008)Google Scholar
  5. 5.
    Crochemore, M., Rytter, W.: Squares, cubes and time-space efficient stringsearching. Algorithmica 13(5), 405–425 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Giraud, M.: Not so many runs in strings. In: Martin-Vide, C. (ed.) 2nd International Conference on Language and Automata Theory and Applications (2008)Google Scholar
  7. 7.
    Iliopoulos, C.S., Moore, D., Smyth, W.F.: A characterization of the squares in a Fibonacci string. Theoret. Comput. Sci. 172(1–2), 281–291 (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kolpakov, R., Kucherov, G.: Finding maximal repetitions in a word in linear time. In: Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, pp. 596–604. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  9. 9.
    Kolpakov, R., Kucherov, G.: On maximal repetitions in words. J. Discret. Algorithms 1(1), 159–186 (2000)MathSciNetGoogle Scholar
  10. 10.
    Lothaire, M.: Applied Combinatorics on Words. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  11. 11.
    Main, M.G.: Detecting leftmost maximal periodicities. Discret. Appl. Math. 25, 145–153 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Main, M.G., Lorentz, R.J.: An O(n log n) algorithm for finding all repetitions in a string. J. Algorithms 5(3), 422–432 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Puglisi, S.J., Simpson, J., Smyth, W.F.: How many runs can a string contain? Personal communication (submitted, 2007)Google Scholar
  14. 14.
    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
  15. 15.
    Rytter, W.: The number of runs in a string. Inf. Comput. 205(9), 1459–1469 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Thue, A.: Uber unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I Math-Nat. Kl. 7, 1–22 (1906)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Maxime Crochemore
    • 1
    • 2
  • Szilárd Zsolt Fazekas
    • 3
  • Costas Iliopoulos
    • 1
  • Inuka Jayasekera
    • 1
  1. 1.King’s CollegeLondonU.K.
  2. 2.Université Paris-EstFrance
  3. 3.Rovira i Virgili UniversityTarragonaSpain

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