Abstract
Domino tilings of finite domains of the plane are used to model dimer systems in statistical physics. In this paper, we study dimer tilings, which generalize domino tilings and are indeed equivalent to perfect matchings of planar graphs. We use height functions, a notion previously introduced by Thurston in [10] for domino tilings, to prove that a dimer tiling of a given domain can be computed using any Single-Source-Shortest-Paths algorithm on a planar graph. We also endow the set of dimers tilings of a given domain with a structure of distributive lattice and show that it can be effectively visited by a simple algorithmical operation called flip.
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References
Bodini, O., Latapy, M.: Generalized Tilings with Height Functions. Morfismos 7 (2003)
Desreux, S., Matamala, M., Rapaport, I., Remila, E.: Domino tiling and related models: space of configurations of domains with holes. Theoret. Comput. Sci. 319, 83–101 (2004)
Fakcharoenphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. In: FOCS 2001, pp. 232–241 (2001)
Fernique, T.: Pavages d’une polycellule. LIRMM Research Report 04002 (2004), Available at: http://www.lirmm.fr/~fernique/info/memoire_mim3.ps.gz
Kasteleyn, P.W.: The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)
Kenyon, R.: The planar dimer model with boundary: a survey. In: Baake, M., Moody, R. (eds.) Directions in mathematical quasicrystals. CRM monograph series. AMS, Providence, RI (2000)
Propp, J.: Lattice structure of orientations of graphs (preprint, 1993), Available at: http://www.math.wisc.edu/~propp/orient.html
Propp, J.: Generating random elements of finite distributive lattices (preprint, 1997), Available at: http://www.math.wisc.edu/~propp/wilf.ps.gz
Thiant, N.: An \(\mathcal{O}(n \log n)\)-algorithm for finding a domino tiling of a plane picture whose number of holes is bounded. Theoret. Comput. Sci. 303, 353–374 (2003)
Thurston, W.P.: Conway’s tiling group. American Mathematical Monthly 97, 757–773 (1990)
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© 2006 Springer-Verlag Berlin Heidelberg
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Bodini, O., Fernique, T. (2006). Planar Dimer Tilings. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds) Computer Science – Theory and Applications. CSR 2006. Lecture Notes in Computer Science, vol 3967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753728_13
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DOI: https://doi.org/10.1007/11753728_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34166-6
Online ISBN: 978-3-540-34168-0
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