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.
This material is based upon work supported by the National Science Foundation under Grant Nos. DMS–0754154 and DMS–1060775.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bean, D.R., Ehrenfeucht, A., McNulty, G.: Avoidable patterns in strings of symbols. Pacific Journal of Mathematics 85, 261–294 (1979)
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)
Blanchet-Sadri, F., Lohr, A., Scott, S.: Computing the partial word avoidability indices of binary patterns. Journal of Discrete Algorithms 23, 113–118 (2013)
Blanchet-Sadri, F., Lohr, A., Scott, S.: Computing the partial word avoidability indices of ternary patterns. Journal of Discrete Algorithms 23, 119–142 (2013)
Blanchet-Sadri, F., Mercaş, R., Simmons, S., Weissenstein, E.: Avoidable binary patterns in partial words. Acta Informatica 48, 25–41 (2011)
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)
Blanchet-Sadri, F., Woodhouse, B.: Strict bounds for pattern avoidance. Theoretical Computer Science 506, 17–28 (2013)
Cassaigne, J.: Motifs évitables et régularités dans les mots. Ph.D. thesis, Paris VI (1994)
Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)
Manea, F., Mercaş, R.: Freeness of partial words. Theoretical Computer Science 389, 265–277 (2007)
Ochem, P.: A generator of morphisms for infinite words. RAIRO-Theoretical Informatics and Applications 40, 427–441 (2006)
Zimin, A.I.: Blocking sets of terms. Mathematics of the USSR-Sbornik 47, 353–364 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Blanchet-Sadri, F., Lohr, A., Simmons, S., Woodhouse, B. (2014). Computing Depths of Patterns. In: Dediu, AH., Martín-Vide, C., Sierra-Rodríguez, JL., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2014. Lecture Notes in Computer Science, vol 8370. Springer, Cham. https://doi.org/10.1007/978-3-319-04921-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-04921-2_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04920-5
Online ISBN: 978-3-319-04921-2
eBook Packages: Computer ScienceComputer Science (R0)