Advertisement

Maximal Unbordered Factors of Random Strings

  • Patrick Hagge CordingEmail author
  • Mathias Bæk Tejs Knudsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9954)

Abstract

A border of a string is a non-empty prefix of the string that is also a suffix of the string, and a string is unbordered if it has no border. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the following: If we pick a string of length n from a fixed alphabet uniformly at random, then the expected length of the maximal unbordered factor is \(n - O(1)\). We prove that this conjecture is true by proving that the expected value is in fact \(n - \varTheta (\sigma ^{-1})\), where \(\sigma \) is the size of the alphabet. We discuss some of the consequences of this theorem.

Keywords

Knowledge Discovery Pattern Match Match Algorithm Database Management Minimal Period 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Duval, J.-P.: Relationship between the period of a finite word and the length of its unbordered segments. Discrete Math. 40(1), 31–44 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ehrenfeucht, A., Silberger, D.: Periodicity and unbordered segments of words. Discrete Math. 26(2), 101–109 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gawrychowski, P., Kucherov, G., Sach, B., Starikovskaya, T.: Computing the longest unbordered substring. In: Iliopoulos, C., Puglisi, S., Yilmaz, E. (eds.) SPIRE 2015. LNCS, vol. 9309, pp. 246–257. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-23826-5_24 CrossRefGoogle Scholar
  4. 4.
    Harju, T., Nowotka, D.: Periodicity and unbordered words: a proof of the extended duval conjecture. J. ACM (JACM) 54(4), 20 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Holub, Š., Nowotka, D.: The ehrenfeucht-silberger problem. J. Comb. Theor. Ser. A 119(3), 668–682 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Holub, Š., Shallit, J.: Periods and borders of random words. In: 33rd Symposium on Theoretical Aspects of Computer Science (2016)Google Scholar
  7. 7.
    Knuth, D.E., Morris Jr., J.H., Pratt, V.R.: Fast pattern matching in strings. SIAM J. Comput. 6(2), 323–350 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kociumaka, T., Radoszewski, J., Rytter, W., Waleń, T.: Efficient data structures for the factor periodicity problem. In: Calderón-Benavides, L., González-Caro, C., Chávez, E., Ziviani, N. (eds.) SPIRE 2012. LNCS, vol. 7608, pp. 284–294. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-34109-0_30 CrossRefGoogle Scholar
  9. 9.
    Kociumaka, T., Radoszewski, J., Rytter, W., Waleń, T.: Internal pattern matching queries in a text and applications. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 532–551. SIAM (2015)Google Scholar
  10. 10.
    Loptev, A., Kucherov, G., Starikovskaya, T.: On maximal unbordered factors. In: Cicalese, F., Porat, E., Vaccaro, U. (eds.) CPM 2015. LNCS, vol. 9133, pp. 343–354. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-19929-0_29 CrossRefGoogle Scholar
  11. 11.
    Morris Jr., J.H., Pratt, V.R.: A linear pattern-matching algorithm (1970)Google Scholar
  12. 12.
    Nielsen, P.T.: A note on bifix-free sequences (corresp.). IEEE Trans. Inf. Theor. 19(5), 704–706 (1973)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Patrick Hagge Cording
    • 1
    Email author
  • Mathias Bæk Tejs Knudsen
    • 2
  1. 1.DTU ComputeTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations