Advertisement

Algorithmica

, Volume 9, Issue 4, pp 382–397 | Cite as

Tiling a polygon with parallelograms

  • Richard Kenyon
Article

Abstract

Given a simple polygon in the plane we devise a quadratic algorithm for determining the existence of, and constructing, a tiling of the polygon with parallelograms. We also show that any two parallelogram tilings can be obtained from one another by a sequence of “rotations,” and give a condition for the uniqueness of such a tiling. Three generalizations of this problem, that of tiling by a fixed set of triangles, a fixed set of trapezoids, or parallelogram tiling for polygonal regions with holes, are shown to be NP-complete.

Key words

Tiling NP-complete 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    R. Berger.The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society, No. 66. AMS, Providence, RI, 1966.Google Scholar
  2. [C]
    B. Chazelle. Triangulating a simple polygon in linear time.Proceedings of the 31st Annual Symposium on the Foundations of Computer Science, 1990, pp. 220–230.Google Scholar
  3. [GJP]
    M. S. Garey, D. S. Johnson, and C. H. Papadimitriou, unpublished results, 1977.Google Scholar
  4. [KS]
    S. Kannan and D. Soroker. Tiling polygons with parallelograms. Preprint, IBM.Google Scholar
  5. [Ka]
    R. M. Karp. Reducibility among combinatorial problems, inComplexity of Computer Computations (R. E. Miller and J. W. Thatcher, eds.). Plenum, New York, 1972, pp. 85–103.Google Scholar
  6. [Ke]
    R. Kenyon. Self-Similar Tilings. Thesis, Princeton University, 1990.Google Scholar
  7. [R]
    J. Robinson. Undecidability and non-periodicity of tilings in the plane.Invent. Math.,12 (1971), 177–209.Google Scholar
  8. [T]
    W. P. Thurston. Conway's tiling groups.Amer. Math. Monthly, October 1990, pp. 757–773.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Richard Kenyon
    • 1
  1. 1.Institut FourierSaint-Martin d'HeresFrance

Personalised recommendations