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)


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.


Knowledge Discovery Pattern Match Match Algorithm Database Management Minimal Period 
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  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

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