An Optimal Algorithm for Detecting Pseudo-squares

  • Srečko Brlek
  • Xavier Provençal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


We consider the problem of determining if a given word, which encodes the boundary of a discrete figure, tiles the plane by translation. These words have been characterized by the Beauquier-Nivat condition, for which we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon.


  1. 1.
    Braquelaire, J.-P., Vialard, A.: Euclidean Paths: A New Representation of Boundary of Discrete Regions. Graphical Models and Image Processing 61(1), 16–43 (1999)zbMATHCrossRefGoogle Scholar
  2. 2.
    Beauquier, D., Nivat, M.: On Translating one Polyomino to Tile the Plane. Discrete Comput. Geom. 6, 575–592 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brlek, S., Labelle, G., Lacasse, A.: A Note on a Result of Daurat and Nivat. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 189–198. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Daurat, A., Nivat, M.: Salient and Reentrant Points of Discrete Sets. In: Del Lungo, A., Di Gesu, V., Kuba, A. (eds.) Proc. International Workshop on Combinatorial Image Analysis (IWCIA 2003), Palermo, Italy, May 14–16, Electronic Notes in Discrete Mathematics, vol. 12, Elsevier Science Publishers, Amsterdam (2003)Google Scholar
  5. 5.
    Freeman, H.: On the Encoding of Arbitrary Geometric Configurations. IRE Trans. Electronic Computer 10, 260–268 (1961)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Freeman, H.: Boundary encoding and processing. In: Lipkin, B.S., Rosenfeld, A. (eds.) Picture Processing and Psychopictorics, pp. 241–266. Academic Press, New York (1970)Google Scholar
  7. 7.
    Gambini, I., Vuillon, L.: An algorithm for deciding if a polyomino tiles the plane by translations, LAMA research report (2003)Google Scholar
  8. 8.
    Gusfield, D., Stoye, J.: Linear time algorithms for finding and representing all the tandem repeats in a string. Journal of Computer and System Sciences 69, 525–546 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gusfield, D.: Algorithms on Strings, Trees and Sequences. Cambridge University Press, Cambridge (1997)zbMATHCrossRefGoogle Scholar
  10. 10.
    Ilie, L.: A note on the number of distinct squares in a word. In: Brlek, S., Reutenauer, C. (eds.) Proc. 5th International Conference on Words, Words 2005, Montreal, Canada, September 13–17, vol. 36, pp. 289–294. Publications du LaCIM (2005)Google Scholar
  11. 11.
    Lothaire, M.: Applied Combinatorics on words. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  12. 12.
    Wijshoff, H.A.G., Van Leeuven, J.: Arbitrary versus periodic storage schemes and tesselations of the plane using one type of polyomino. Inform. Control 62, 1–25 (1984)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Srečko Brlek
    • 1
  • Xavier Provençal
    • 1
  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontréalCanada

Personalised recommendations