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Computing Depths of Patterns

  • Francine Blanchet-Sadri
  • Andrew Lohr
  • Sean Simmons
  • Brent Woodhouse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)

Abstract

Pattern avoidance is an important research topic in combinatorics on words which dates back to Thue’s construction of an infinite word over three letters that avoids squares, i.e., a sequence with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. A more general form of pattern avoidance has recently emerged to allow for undefined positions in sequences. New concepts on patterns such as depth have been introduced and a number of questions have been raised, some of them we answer. In the process, we prove a strict bound on the number of square occurrences in an unavoidable pattern, and consequently, any pattern with more square occurrences than distinct variables is avoidable over three letters. We also prove a strict bound on the length of an avoidable pattern with at least four distinct variables. We finally provide an algorithm that determines whether a given pattern is of bounded depth, and if so, computes its depth.

Keywords

Depth Function Distinct Variable Alphabet Size Partial Word Pattern Avoidance 
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.

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References

  1. 1.
    Bean, D.R., Ehrenfeucht, A., McNulty, G.: Avoidable patterns in strings of symbols. Pacific Journal of Mathematics 85, 261–294 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Blanchet-Sadri, F., Black, K., Zemke, A.: Unary pattern avoidance in partial words dense with holes. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 155–166. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Blanchet-Sadri, F., Lohr, A., Scott, S.: Computing the partial word avoidability indices of binary patterns. Journal of Discrete Algorithms 23, 113–118 (2013)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Blanchet-Sadri, F., Lohr, A., Scott, S.: Computing the partial word avoidability indices of ternary patterns. Journal of Discrete Algorithms 23, 119–142 (2013)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Blanchet-Sadri, F., Mercaş, R., Simmons, S., Weissenstein, E.: Avoidable binary patterns in partial words. Acta Informatica 48, 25–41 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blanchet-Sadri, F., Simmons, S.: Abelian pattern avoidance in partial words. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 210–221. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Blanchet-Sadri, F., Woodhouse, B.: Strict bounds for pattern avoidance. Theoretical Computer Science 506, 17–28 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cassaigne, J.: Motifs évitables et régularités dans les mots. Ph.D. thesis, Paris VI (1994)Google Scholar
  9. 9.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Manea, F., Mercaş, R.: Freeness of partial words. Theoretical Computer Science 389, 265–277 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ochem, P.: A generator of morphisms for infinite words. RAIRO-Theoretical Informatics and Applications 40, 427–441 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Zimin, A.I.: Blocking sets of terms. Mathematics of the USSR-Sbornik 47, 353–364 (1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Francine Blanchet-Sadri
    • 1
  • Andrew Lohr
    • 2
  • Sean Simmons
    • 3
  • Brent Woodhouse
    • 4
  1. 1.Department of Computer ScienceUniversity of North CarolinaGreensboroUSA
  2. 2.Department of Mathematics, Mathematics BuildingUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Department of MathematicsPurdue UniversityWest LafayetteUSA

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