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On the Arithmetics of Discrete Figures

  • Alexandre Blondin Massé
  • Amadou Makhtar Tall
  • Hugo Tremblay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)

Abstract

Discrete figures (or polyominoes) are fundamental objects in combinatorics and discrete geometry, having been studied in many contexts, ranging from game theory to tiling problems. In 2008, Provençal introduced the concept of prime and composed polyominoes, which arises naturally from a composition operator acting on these discrete figures. Our goal is to study further polyomino composition and, in particular, factorization of polyominoes as a product of prime ones. We provide a polynomial time (with respect to the perimeter of the polyomino) algorithm that allows one to compute such a factorization. As a consequence, primality of polyominoes can be decided in polynomial time.

Keywords

Discrete figures polyominoes boundary words primality tiling morphism 

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References

  1. 1.
    Beauquier, D., Nivat, M.: On translating one polyomino to tile the plane. Discrete Comput. Geom. 6, 575–592 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brlek, S., Provençal, X.: On the problem of deciding if a polyomino tiles the plane by translation. In: Holub, J., Žďárek, J. (eds.) Proceedings of the Prague Stringology Conference 2006, Czech Technical University in Prague, Prague, Czech Republic, August 28-30, pp. 65–76 (2006) ISBN80-01-03533-6Google Scholar
  3. 3.
    Brlek, S., Frosini, A., Rinaldi, S., Vuillon, L.: Tilings by translation: enumeration by a rational language approach. Electronic Journal of Combinatorics 13, 15 (2006)MathSciNetGoogle Scholar
  4. 4.
    Gambini, I., Vuillon, L.: An algorithm for deciding if a polyomino tiles the plane by translations. Theoret. Informatics Appl. 41, 147–155 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Golomb, S.W.: Tiling with sets of polyominoes. Journal of Combinatorial Theory 9(1), 60–71 (1970), http://www.sciencedirect.com/science/article/pii/S0021980070800552 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Golomb, S.W.: Polyominoes: Puzzles, Patterns, Problems, and Packings. Princeton Academic Press, Princeton (1996)Google Scholar
  7. 7.
    Knuth, D.E.: Dancing links (2000), http://arxiv.org/abs/cs/0011047
  8. 8.
  9. 9.
    Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    Blondin-Massé, A., Brlek, S., Garon, A., Labbé, S.: Christoffel and Fibonacci tiles. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 67–78. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Blondin Massé, A., Garon, A., Labbé, S.: Combinatorial properties of double square tiles. Theoretical Computer Science (2012), http://www.sciencedirect.com/science/article/pii/S0304397512009723, http://www.sciencedirect.com/science/article/pii/S0304397512009723
  12. 12.
    Massé, A.B., Frosini, A., Rinaldi, S., Vuillon, L.: On the shape of permutomino tiles. Discrete Applied Mathematics 161(15), 2316–2327 (2013), http://www.sciencedirect.com/science/article/pii/S0166218X12003344; advances in Discrete Geometry: 16th International Conference on Discrete Geometry for Computer ImageryGoogle Scholar
  13. 13.
    Moore, C., Michael, J.: Hard tiling problems with simple tiles (2000), http://arxiv.org/abs/math/0003039
  14. 14.
    Polyá, G.: On the number of certain lattice polygons. Journal of Combinatorial Theory 6(1), 102–105 (1969), http://www.sciencedirect.com/science/article/pii/S0021980069801134 CrossRefzbMATHGoogle Scholar
  15. 15.
    Provençal, X.: Combinatoire des mots, géométrie discrète et pavages. Ph.D. thesis, D1715, Université du Québec à Montréal (2008)Google Scholar
  16. 16.
    Wijshoff, H., van Leeuven, J.: Arbitrary versus periodic storage schemes and tesselations of the plane using one type of polyomino. Inform. Control 62, 1–25 (1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alexandre Blondin Massé
    • 1
    • 2
  • Amadou Makhtar Tall
    • 1
  • Hugo Tremblay
    • 1
    • 2
  1. 1.Laboratoire d’informatique formelleUniversité du Québec à ChicoutimiChicoutimiCanada
  2. 2.Laboratoire de combinatoire et d’informatique mathématiqueUniversité du Québec à MontréalMontralCanada

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