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Optimal Partial Tiling of Manhattan Polyominoes

  • Olivier Bodini
  • Jérémie Lumbroso
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)

Abstract

Finding an efficient optimal partial tiling algorithm is still an open problem. We have worked on a special case, the tiling of Manhattan polyominoes with dominoes, for which we give an algorithm linear in the number of columns. Some techniques are borrowed from traditional graph optimisation problems.

Keywords

Bipartite Graph Greedy Algorithm Combinatorial Group Theory Tiling Problem Domino Tiling 
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.

References

  1. 1.
    Bodini, O., Latapy, M.: Generalized Tilings with Height Functions. Morfismos 7 3 (2003)Google Scholar
  2. 2.
    Bougé, L., Cosnard, M.: Recouvrement d’une pièce trapézoidale par des dominos, C. R. Académie des Sciences Paris 315, Série I, pp. 221–226 (1992)Google Scholar
  3. 3.
    Bodini, O., Fernique, T.: Planar Tilings. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 104–113. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Conway, J.H., Lagarias, J.C.: Tiling with polyominoes and combinatorial group theory. JCT Series A 53, pp. 183–208 (1990)Google Scholar
  5. 5.
    Ford, L.R., Fulkerson, D.R.: Maximal Flow through a Network. Canadian Journal of Mathematics, 399 (1956)Google Scholar
  6. 6.
    Fournier, J.C.: Tiling pictures of the plane with dominoes. Discrete Mathematics 165/166, 313–320 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18(5), 1013–1036 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ito, K.: Domino tilings on planar regions. J.C.T. Series A 75, pp. 173–186 (1996)Google Scholar
  10. 10.
    Kenyon, R.: A note on tiling with integer-sided rectangles. JCT Series A 74, pp. 321–332 (1996)Google Scholar
  11. 11.
    Rémila, E.: The lattice structure of the set of domino tilings of a polygon. Theoretical Computer Science 322(2), 409–422 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Thiant, N.: An O (n log n)-algorithm for finding a domino tiling of a plane picture whose number of holes is bounded. Theoretical Computer Science 303(2-3), 353–374 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Thurston, W.P.: Conway’s tiling groups. American Mathematics Monthly 95, 757–773 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Olivier Bodini
    • 1
  • Jérémie Lumbroso
    • 1
  1. 1.LIP6, UMR 7606, CALSCI departementUniversité Paris 6 / UPMCParis cedex 05France

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