Advertisement

Periods in Partial Words: An Algorithm

  • Francine Blanchet-Sadri
  • Travis Mandel
  • Gautam Sisodia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

Partial words are finite sequences over a finite alphabet that may contain some holes. A variant of the celebrated Fine-Wilf theorem shows the existence of a bound L = L(h,p,q) such that if a partial word of length at least L with h holes has periods p and q, then it has period \(\gcd(p,q)\). In this paper, we associate a graph with each p- and q-periodic word, and study two types of vertex connectivity on such a graph: modified degree connectivity and r-set connectivity where \(r = q \bmod{p}\). As a result, we give an algorithm for computing L(h, p, q) in the general case.

Keywords

Optimal Length Repeated Pattern Closed Formula Vertex Connectivity Partial Word 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Choffrut, C., Karhumäki, J.: Combinatorics of Words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 329–438. Springer, Berlin (1997)CrossRefGoogle Scholar
  2. 2.
    Smyth, W.F.: Computing Patterns in Strings. Pearson, Addison-Wesley (2003)Google Scholar
  3. 3.
    Fine, N.J., Wilf, H.S.: Uniqueness theorems for periodic functions. Proceedings of the American Mathematical Society 16, 109–114 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Constantinescu, S., Ilie, L.: Generalised Fine and Wilf’s theorem for arbitrary number of periods. Theoretical Computer Science 339, 49–60 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berstel, J., Boasson, L.: Partial words and a theorem of Fine and Wilf. Theoretical Computer Science 218, 135–141 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchet-Sadri, F.: Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008)zbMATHGoogle Scholar
  7. 7.
    Blanchet-Sadri, F., Bal, D., Sisodia, G.: Graph connectivity, partial words, and a theorem of Fine and Wilf. Information and Computation 206(5), 676–693 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Halava, V., Harju, T., Kärki, T.: Interaction properties of relational periods. Discrete Mathematics and Theoretetical Computer Science 10, 87–112 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Shur, A.M., Gamzova, Y.V.: Partial words and the interaction property of periods. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 68(2), 191–214 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Shur, A.M., Konovalova, Y.V.: On the Periods of Partial Words. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 657–665. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Smyth, W.F., Wang, S.: A new approach to the periodicity lemma on strings with holes. Theoretical Computer Science 410, 4295–4302 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Francine Blanchet-Sadri
    • 1
  • Travis Mandel
    • 2
  • Gautam Sisodia
    • 3
  1. 1.Department of Computer ScienceUniversity of North CarolinaGreensboroUSA
  2. 2.Department of MathematicsThe University of TexasAustinUSA
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations