Abstract
Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston’s height function approach to a nearly linear time in the perimeter.
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Beauquier, D., Nivat, M., Rémila, É., Robson, M.: Tiling figures of the plane with two bars. Comput. Geom. 5, 1–25 (1995)
Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66, 72 (1966)
Bodini, O., Fernique, T., Rémila, É.: A characterization of flip-accessibility for rhombus tilings of the whole plane. Inform. Comput. 206, 1065–1073 (2008)
Chaboud, T.: Domino tiling in planar graphs with regular and bipartite dual. Theor. Comput. Sci. 159, 137–142 (1996)
Conway, J.H., Lagarias, J.C.: Tilings with polyominoes and combinatorial group theory. J. Comb. Theory Ser. A 53, 183–208 (1990)
Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15, 317–340 (1986)
Fournier, J.C.: Pavage des figures planes sans trous par des dominos (in French). Theor. Comput. Sci. 159, 105–128 (1996)
Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)
Goldreich, O.: Property testing in massive graphs. In: Abello, J., Pardalos, P.M., Resende, M. (eds.) Handbook of Massive Data Sets, pp. 123–147. Kluwer Academic Publishers, Dordrecht (2002)
Golomb, S.: Polyominoes. Scribners, New York (1965)
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)
Ito, K.: Domino tilings on orientable surfaces. J. Comb. Theory Ser. A 84, 1–8 (1998)
Kenyon, C., Kenyon, R.: Tiling a polygon with rectangles. In: Proceedings of 33rd FOCS, Pittsburgh, PA, pp. 610–619 (1992)
Kenyon, R.: The planar dimer model with boundary: a survey. In: Baake, M., Moody, R. (eds.) Directions in Mathematical Quasicrystals, pp. 307–328. American Mathematical Society, Providence, RI (2000)
Kenyon, R.: An introduction to the dimer model. In: ICTP Lecture Notes Series XVII. ICTP, Trieste (2004)
Kenyon, R.: Lectures on Dimers. arXiv preprint http://arxiv.org/abs/0910.3129 (2009)
Korn, M.: Geometric and algebraic properties of polyomino tilings. MIT Ph.D. thesis (2004). http://dspace.mit.edu/handle/1721.1/16628
Linde, J., Moore, C., Nordahl, M.G.: An \(n\)-dimensional generalization of the rhombus tiling. In: Proceedings on discrete models: combinatorics, computation, and geometry, pp. 23–42. MIMD, Paris (2001)
Levin, L.: Universal sorting problems. Probl. Inf. Transm. 9, 265–266 (1973)
Lovász, L., Plummer, M.D.: Matching Theory. American Mathematical Society, Providence, RI (2009)
Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31, 167–192 (2001)
Miller, G.L., Naor, J.: Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24, 1002–1017 (1995)
Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26, 573–590 (2001)
Mozes, S., Wulff-Nilsen, C.: Shortest paths in planar graphs with real lengths in \(O(n\log ^2n/\log \log n)\) time. In: Proceedings of 18th ESA, pp. 206–217. Springer, Berlin (2010)
Pak, I.: Tile invariants: new horizons. Theor. Comput. Sci. 303, 303–331 (2003)
Pak, I., Yang, J.: Tiling simply connected regions with rectangles. J. Comb. Theory Ser. A 120, 1804–1816 (2013)
Pak, I., Yang, J.: The complexity of generalized domino tilings. Electron. J. Comb. 20(4), 12–23 (2013)
Rémila, É.: Tiling groups: new applications in the triangular lattice. Discrete Comput. Geom. 20, 189–204 (1998)
Rémila, É.: Tiling a polygon with two kinds of rectangles. Discrete Comput. Geom. 34, 313–330 (2005)
Rubinfeld, R., Shapira, A.: Sublinear time algorithms. SIAM J. Discrete Math. 25, 1562–1588 (2011)
Saldanha, N.C., Tomei, C., Casarin, M.A., Romualdo, D.: Spaces of domino tilings. Discrete Comput. Geom. 14, 207–233 (1995)
Tassy, M.: Tiling by bars. Ph.D. Thesis, Brown University, Providence, RI (2014)
Thiant, N.: An \(O(n\log n)\)-algorithm for finding a domino tiling of a plane picture whose number of holes is bounded. Theor. Comput. Sci. 303, 353–374 (2003)
Thurston, W.P.: Conway’s tiling groups. Am. Math. Mon. 97, 757–773 (1990)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)
Valiant, L.G.: Completeness classes in algebra. In: Proceedings of 11th STOC, pp. 249–261. ACM, New York (1979)
van Emde Boas, P.: The convenience of tilings. In: Sorbi, A. (ed.) Complexity, Logic, and Recursion Theory, pp. 331–363. Dekker, New York (1997)
Acknowledgments
We are very grateful to Scott Garrabrant and Yahav Nussbaum for interesting discussions and helpful remarks. The first author was partially supported by the NSF.
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Pak, I., Sheffer, A. & Tassy, M. Fast Domino Tileability. Discrete Comput Geom 56, 377–394 (2016). https://doi.org/10.1007/s00454-016-9807-1
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DOI: https://doi.org/10.1007/s00454-016-9807-1