# Fast Domino Tileability

```@article{Pak2016FastDT,
title={Fast Domino Tileability},
author={Igor Pak and Adam Sheffer and Martin Tassy},
journal={Discrete \& Computational Geometry},
year={2016},
volume={56},
pages={377-394}
}```
• Published 2 July 2015
• Mathematics
• Discrete & Computational Geometry
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.
10 Citations
Tiling with Squares and Packing Dominos in Polynomial Time
The standard, compact way to represent a polygon is by listing the coordinates of the corners in binary, and thus the first polynomial time algorithms for the problems are presented, including a simple \$O(n\log n) \$ algorithm for tiling with squares, and a more involved \$O (n^4)\$ algorithm for packing and tiled with dominos.
MIXING PROPERTIES OF COLORINGS OF THE Z LATTICE
• Mathematics
• 2019
We study and classify proper q-colorings of the Zd lattice, identifying three regimes where different combinatorial behavior holds: (1) When q ≤ d + 1, there exist frozen colorings, that is, proper
A quenched variational principle for discrete random maps
• Mathematics
• 2017
We study the variational principle for discrete height functions (or equivalently domino tilings) where the underlying measure is perturbed by a random field. We show that the variational principle
Mixing properties of colourings of the ℤd lattice
• Mathematics
Comb. Probab. Comput.
• 2021
A strong list-colours property is proved which implies that, when \$q\ge d+2\$ , any proper q-colouring of the boundary of a box of side length \$n \ge d-2\$ can be extended to a properq-coloured of the entire box.
Kirszbraun-type theorems for graphs
• Mathematics
J. Comb. Theory, Ser. B
• 2019
Mixing properties of colorings of the \$\mathbb{Z}^d\$ lattice
• Mathematics
• 2019
We study and classify proper \$q\$-colorings of the \$\mathbb Z^d\$ lattice, identifying three regimes where different combinatorial behavior holds: (1) When \$q\le d+1\$, there exist frozen colorings,
Borel subsystems and ergodic universality for compact Zd ‐systems via specification and beyond
• Mathematics
Proceedings of the London Mathematical Society
• 2021
A Borel Zd dynamical system (X,S) is ‘almost Borel universal’ if any free Borel Zd dynamical system (Y,T) of strictly lower entropy is isomorphic to a Borel subsystem of (X,S) , after removing a null
A variational principle for a non-integrable model
• Mathematics
Probability Theory and Related Fields
• 2020
We show the existence of a variational principle for graph homomorphisms from \$\$\mathbb {Z}^m\$\$ Z m to a d -regular tree. The technique is based on a discrete Kirszbraun theorem and a concentration
Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings
• Mathematics
Comb. Probab. Comput.
• 2022
Borel subsystems and ergodic universality for compact \$\mathbb Z^d\$-systems via specification and beyond
• Mathematics
• 2019
A Borel system \$(X,S)\$ is `almost Borel universal' if any free Borel dynamical system \$(Y,T)\$ of strictly lower entropy is isomorphic to a Borel subsystem of \$(X,S)\$, after removing a null set. We

## References

SHOWING 1-10 OF 36 REFERENCES
The Complexity of Generalized Domino Tilings
• Mathematics
Electron. J. Comb.
• 2013
A variety of hardness results are proved (both NP- and #P-completeness) for different generalizations of dominoes in three and higher dimensions.
Hard Tiling Problems with Simple Tiles
• Mathematics
Discret. Comput. Geom.
• 2001
This work shows that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs, and shows NP-completeness for the domino and straight tromino for general areas on the cubic lattice, and for simply connected regions on the four-dimensional hypercubic lattice.
Geometric and algebraic properties of polyomino tilings
In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of
Spaces of domino tilings
• Mathematics
Discret. Comput. Geom.
• 1995
A criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph, which is a CW-complex whose connected components are homotopically equivalent to points or circles.
Tiling a Polygon with Two Kinds of Rectangles
A quadratic time algorithm is given which, given a polygon F as input, produces a tiling of F with translated copies of the authors' rectangles (or indicates that there is no tiling), and it is proved that any pair oftilings can be linked by a sequence of local transformations of tilings, called flips.
Sublinear Time Algorithms
• Computer Science
SIAM J. Discret. Math.
• 2011
This work discusses the types of answers that one can hope to achieve in sublinear time algorithms, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input.
Conway's tiling groups
John Conway discovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given
Tiling with polyominoes and combinatorial group theory
• Mathematics
J. Comb. Theory, Ser. A
• 1990
An n-Dimensional Generalization of the Rhombus Tiling
• Mathematics
DM-CCG
• 2001
It is shown that the rhombus tiling can be generalized to n-dimensional tiles for any \$n ≥ 3, and it is conjecture that a certain local move is ergodic, and conjecture that it has a mixing time of O(L^{n+2} log L) on regions of size \$L\$.