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.
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R. Berger.The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society, No. 66. AMS, Providence, RI, 1966.
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.
M. S. Garey, D. S. Johnson, and C. H. Papadimitriou, unpublished results, 1977.
S. Kannan and D. Soroker. Tiling polygons with parallelograms. Preprint, IBM.
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.
R. Kenyon. Self-Similar Tilings. Thesis, Princeton University, 1990.
J. Robinson. Undecidability and non-periodicity of tilings in the plane.Invent. Math.,12 (1971), 177–209.
W. P. Thurston. Conway's tiling groups.Amer. Math. Monthly, October 1990, pp. 757–773.
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Communicated by Jean-Daniel Bolssonnat.
Parts of this paper originally appeared in a technical report for the conference “Journées de Géometrie Algorithmique” at Sophia-Antipolis, France, under the same title. The author was supported by a Chateaubriand Fellowship and this work was partially completed while the author was at Princeton University, supported by an IBM graduate fellowship.
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Kenyon, R. Tiling a polygon with parallelograms. Algorithmica 9, 382–397 (1993). https://doi.org/10.1007/BF01228510
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DOI: https://doi.org/10.1007/BF01228510