Z-Tilings of Polyominoes and Standard Basis

  • Olivier Bodini
  • Bertrand Nouvel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)


In this paper, we prove that for every set E of polyominoes (for us, a polyomino is a finite union of unit squares of a square lattice), we have an algorithm which decides in polynomial time, for every polyomino P, whether P has or not a ℤ-tiling (signed tiling) by translated copies of elements of E. Moreover, if P is ℤ-tilable, we can build a ℤ-tiling of P. We use for this the theory of standard basis on ℤ[X 1,...,X n ]. In application, we algorithmically extend results of Conway and Lagarias on ℤ-tiling problems.


Standard Basis Hexagonal Lattice Generalize Coloring Minimal Polynomial Zero Divisor 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barnes, F.W.: Best packing of rods into boxes. Discrete Mathematics 142, 271–275 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bodini, O.: Pavage des polyominos et Bases de Grobner, Rapport de recherche No RR2001-51, LIP (2001)Google Scholar
  3. 3.
    Buchberger, B.: Introduction to Grobner basis. In: Logic of computation (Marktoberdorf 95). NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 157, pp. 35–66. Springer, Berlin (1997)Google Scholar
  4. 4.
    Conway, J.H., Lagarias, J.C.: Tiling with polyominoes and combinatorial group theory. J.C.T. Series A 53, 183–208 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cox, D., Little, J., O’Shea, D.: Ideals,varieties and algorithms, 2nd edn. Undergraduate Text in Mathematics, vol. XV, p. 536. Springer, New York (1997)Google Scholar
  6. 6.
    Faugére, J.-C.: A new efficient algorithm for computing Grobner basis. Journal of Pure and Applied Algebra 139, 61–88 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Golomb, S.W.: Tiling with polyominoes. J.C.T. Series A 1, 280–296 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Golomb, S.W.: Polyominoes which tile rectangles. J.C.T. Series A 51, 117–124 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kenyon, R.: Sur la dynamique, la combinatoire et la statistique des pavages, Habilitation (1999)Google Scholar
  10. 10.
    Klarner, D.A.: Packing a rectangle with congruent n-ominoes. J.C.T. Series A 7, 107–115 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Thurston, W.P.: Conway’s tiling groups. Amer. Math. Monthly 97(8), 757–773 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Olivier Bodini
    • 1
  • Bertrand Nouvel
    • 2
  1. 1.LIRMMMontpellier Cedex 5France
  2. 2.LIP UMR 5668 CNRS-INRIA-ENS Lyon-Univ. Lyon 1Lyon Cedex 07France

Personalised recommendations