Lozenge Tiling by Computing Distances

TL;DR

A graph-theoretic overlay based on directed cuts and systems of difference constraints that complements Thurston's theory of lozenge tilings and makes its algorithmic structure explicit is introduced.

Abstract

The Calisson puzzle is a tiling puzzle in which one must tile a triangular grid inside a hexagon with lozenges, under the constraint that certain prescribed edges remain tile boundaries and that adjacent lozenges along these edges have different orientations. We present the first polynomial-time algorithm for this problem, with cubic running time. This algorithm, called the advancing surface algorithm, can be executed in a simple and intuitive way, even by hand with a pencil and an eraser. Its apparent simplicity conceals a deeper algorithmic reinterpretation of the classical ideas of John Conway and William Thurston, revisited here from a theoretical computer science perspective. We introduce a graph-theoretic overlay based on directed cuts and systems of difference constraints that complements Thurston's theory of lozenge tilings and makes its algorithmic structure explicit. In Thurston's approach, lozenge tilings are lifted to monotone stepped surfaces in the three-dimensional cubic lattice and projected back to the plane using height functions, reducing tilability to the computation of heights. We show that selecting a monotone surface corresponds to selecting a directed cut in a periodic directed graph, while height functions arise as solutions of a system of difference constraints. In this formulation, a region is tilable if and only if the associated weighted directed graph contains no cycle of strictly negative weight. This additional graph layer shows that the Bellman-Ford algorithm suffices to decide feasibility and compute solutions. In particular, our framework allows one to decide whether the infinite triangular grid can be tiled while respecting a finite set of prescribed local constraints, even in the absence of boundary conditions.

Figures (20)

• Figure 1: The rules of the puzzle (Courtesy of Olivier Longuet's https://mathix.org/calisson/blog/blog): we give ourselves a set of edges, as drawn in in the top left-hand corner. The goal is to tile the hexagon with lozenges in such a way that the edges given as input are adjacent to two lozenges of different colors.
• Figure 2: Instances of the Calisson puzzle. The instance of size $n=6$ is solved in Fig. \ref{['ads']}.
• Figure 3: The advancing surface algorithm for solving an instance of the Calisson puzzle. We start from a tiling looking as the surface of an empty cube and make it advance little by little by adding small cubes in order to satisfy new constraints. It leads sometimes to loose a previously satisfied constraint but it is part of the algorithm.
• Figure 4: Solving a lozenge tiling instance with non-overlapping and saliency constraints through distance computation. The first step builds a weighted directed graph, with weights $+1$ for blue edges, $0$ for brown edges, and $-1$ for black and red edges. The second step computes shortest-path distances from an arbitrary source vertex $s$. If the graph contains a cycle of strictly negative total weight, the distance constraints are infeasible and the tiling instance admits no solution. Otherwise, as shown here, a tiling is recovered by connecting adjacent vertices whose distance to $s$ differ by exactly $1$.
• Figure 5: The tiling problems that we solve. The left image is an instance of the problem that we denote Tiling$(R,X_1,X_2)$. Given the finite simply connected region $R$ and two sets of edges $X_1$ (the green edges) and $X_2$ (the orange edges), the problem is to tile the region $R$ without overlapping the green and orange edges (non overlapping constraint). The lozenges adjacent to orange edges also must have different orientation (saliency constraint). The right image illustrates the toy problem Tiling$(\triangle ,X_1,X_2)$ where the region $R$ to be tiled is the whole triangular grid denoted $\triangle$ and thus has no boundary. The main contribution of the paper shows how to solve these problems through a system of difference constraints and thus with distance computations.

Theorems & Definitions (12)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Corollary 5
• Theorem 6: Thurston
• Remark 7
• Claim 8
• Proposition 9
• Theorem 9