Lozenge Tiling by Computing Distances

The triangular grid is denoted $\mathrm{△}$. We use the letter $R$ to denote a region of the triangular grid. Formally, $R$ is a simplicial complex whose sets of triangles, edges, and vertices are respectively denoted by $R^2$, $R^1$, and $R^0$. Motivated by the Calisson puzzle, particular attention is paid to regular hexagonal regions. The regular hexagon of size $n$ is denoted by ${\mathrm{⬡}}_n$.

The results presented in this paper do not require the region $R$ to be compact. However, we assume that $R$ is simply connected (i.e connected with no holes). In particular, we consider instances where the region $R$ is the entire infinite triangular grid, denoted $R=\mathrm{△}$, together with a finite set of input edges. The boundary of $R$ is denoted by $\partial R$.

Our results can also be extended to regions whose boundary passes several times through the same vertices or edges. As long as no saliency constraint is imposed on such shared edges, this can be handled by duplicating boundary vertices and edges, as illustrated in Fig. 8. For the sake of simplicity, we omit these cases from the formal statements and proofs.

We consider two types of constraints on edges of the triangular grid. The first set, denoted $X_1$, consists of edges that must not be overlapped by any lozenge. This is the non overlapping constraint. The second set, denoted $X_2$, consists of edges that must not be overlapped and whose two adjacent lozenges must have distinct orientations. This additional requirement is the saliency constraint.

This leads to the following general tiling problem:

Tiling$(R,X_1,X_2)$

• $\mathrm{■}$

  Input: A region $R\subseteq {\mathrm{△}}^2$ and a set $X\subseteq {\mathrm{△}}^1$ of edges of the triangular grid, partitioned into two subsets $X=X_1\cup X_2$.

• $\mathrm{■}$

  Output: A lozenge tiling of the region $R$ such that (i) no edge of $X$ is overlapped by a lozenge, and (ii) for every edge in $X_2$, the two adjacent lozenges have different orientations.

A Calisson puzzle with a set of interior saliency constraints $X_2$ corresponds to the instance Tiling$({\mathrm{⬡}}_n,\mathrm{\varnothing },X_2)$.

The advancing surface algorithm is illustrated in Figs. 3 and 18 on instances of the Calisson puzzle Tiling$({\mathrm{⬡}}_n,\mathrm{\varnothing },X_2)$. It fills an initially empty cube of size $n\times n\times n$ by progressively adding unit cubes, each time adding as few cubes as possible, so as to satisfy the non overlapping and saliency constraints as a simple graph traversal algorithm. It can be easily extended to bounded, simply connected regions of the triangular grid. This extension only requires to add a preprocessing step that computes the minimal and maximal tilings (for instance by running classical Thurston’s algorithm). The advancing surface algorithm then progressively adds unit cubes in the volume between these two extremal tilings. The algorithm and its analysis are detailed in Subsection 5.2.

In the special case of the Calisson puzzle, the region is the hexagon ${\mathrm{⬡}}_n$. Then the two extremal tilings are straightforward (the projections of the surface of an empty and a full $n\times n\times n$ cube).

The theorem 1 and Corollary 2 are proved in Subsection 5.2 after the introduction of a path of consecutive reductions explaining why a so simple algorithm works.

The main contribution of this paper is the completion of Thurston’s theory of lozenge tilings with a graph-theoretical layer. This additional structure allows us to translate the generic tiling problem Tiling$(R,X_1,X_2)$ into an equivalent system of difference constraints. This system is induced by a weighted directed graph, denoted $DC(R,X_1,X_2)$ and defined as follows:

A system of difference constraints induced by a weighted directed graph is the assignation problem of a value $h(v)$ to each vertex $v$ such that, for every directed edge $u\to v$ with weight $w$, the inequality $h(v)-h(u)\le w$ is satisfied (see Fig. 10 or reference CLRS [2] section 24.4).

We claim that the tiling instance Tiling$(R,X_1,X_2)$ is equivalent to the system of difference constraints induced by $DC(R,X_1,X_2)$.

The tiling is obtained from the integer-valued function $h$ by linking the adjacent vertices of $R$ whose value $f$ differ from $1$ (as illustrated in Fig. 4).

Classically, systems of difference constraints are solved by shortest-path algorithms such as Bellman–Ford [2]. If the graph contains a directed cycle of strictly negative total weight, then the system has no feasible solution. Otherwise, shortest-path distances from an arbitrary source yield a valid solution of the constraints.

This dichotomy is illustrated in Fig. 11.

As a direct consequence of Theorem 3, we obtain the following corollary.

We conclude this section by considering the decision problem of tiling the entire triangular grid $\mathrm{△}$ with a finite set of constraints $X_1$ and $X_2$.

Due to the absence of boundary conditions and the infinite number of tiles, this last problem cannot be addressed by classical approaches such as matching-based methods, Thurston’s original algorithm, SAT encodings, or maximum independent set formulations. The corollary 5 illustrates that the difference constraints formulation induced by $DC(R,X_1,X_2)$ is not a mere technical rewriting of Thurston’s theory, but rather an essential complement that clarifies the algorithmic theory of lozenge tilings and enables the solution of tiling problems beyond the reach of standard methods. Due to lack of space, we omit its proof and report the reader to the full Arxiv version of the paper.

The primary unit cube is ${[0,1]}^3$. The cubes of the cubic grid are denoted $(x,y,z)+{[0,1]}^3$, with $(x,y,z)\in {\mathbb{Z}}^3$ (see Fig. 12). The sets of cubes, faces, edges, and vertices of the cubic grid are respectively denoted ${\mathrm{\square }}^3$, ${\mathrm{\square }}^2$, ${\mathrm{\square }}^1$, and ${\mathrm{\square }}^0$. Throughout the paper, we follow the convention that a superscript $k\in \{0,1,2,3\}$ indicates the dimension of the objects in the corresponding set.

The infinite triangular grid $\mathrm{△}$ is obtained by projecting the edges of ${\mathrm{\square }}^1$ using a map $\phi$. More precisely, $\phi$ denotes the projection of the three-dimensional space ${ℝ}^3$ onto a plane $H$ of equation $x+y+z=h$, along the direction $\mathrm{𝟙}=(1,1,1)$. The plane $H$ is naturally decomposed into a simplicial complex $\mathrm{△}={\mathrm{△}}^2\cup {\mathrm{△}}^1\cup {\mathrm{△}}^0$ consisting of triangles, edges, and vertices.

Any vertex of ${\mathrm{△}}^0$ is incident to six edges. Their directions are given by $\phi (1,0,0)$, $\phi (0,1,0)$, $\phi (0,0,1)$, and their opposites. Rather than using two-dimensional coordinates in the plane $H$, we use homogeneous coordinates: a point of $H$ is represented as $\phi (x,y,z)$ with $(x,y,z)\in {ℝ}^3$. Clearly, $\phi (x,y,z)=\phi (x+k,y+k,z+k)$ for any $k\in ℝ$. Adding such a constant changes the height in the $\mathrm{𝟙}$ direction without affecting the projection. This notion of height was introduced by Thurston. The height of $(x,y,z)$ is simply $x+y+z$.

Our two-dimensional tiles are lozenges. A lozenge is defined as the projection under $\phi$ of a two-dimensional face of the cubic grid $\mathrm{\square }$. Since the faces of $\mathrm{\square }$ have three possible orientations, there are exactly three types of lozenges. Blue, red, and yellow lozenges are the projections of square faces with normal directions $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.

With our notation, each cube $(x,y,z)+{[0,1]}^3$ of ${\mathrm{\square }}^3$ is identified with its base point $(x,y,z)$, yielding a natural one-to-one correspondence between ${\mathrm{\square }}^3$ and ${\mathbb{Z}}^3$. We define ${\mathrm{\square }}_R^3\subset {\mathrm{\square }}^3$ as the set of cubes whose base points project into the region $R$, that is, such that $\phi (x,y,z)\in R$. Equivalently, ${\mathrm{\square }}_R^3$ consists of infinite stacks of cubes ${\phi }^{-1}(\phi (x,y,z))$ above the vertices $\phi (x,y,z)\in R$ in the direction $\mathrm{𝟙}$.

We endow ${\mathrm{\square }}_R^3$ with a directed graph structure, called the ascendant graph and denoted ${\mathscr{H}}_R=({\mathrm{\square }}_R^3,{\wedge }_R)$. There is a directed edge $(x,y,z)+{[0,1]}^3\to (x^{\prime },y^{\prime },z^{\prime })+{[0,1]}^3$ in ${\wedge }_R$ if and only if the following two conditions hold:

1. (a)

   the cube $(x^{\prime },y^{\prime },z^{\prime })+{[0,1]}^3$ is one of the three cubes $(x+1,y,z)+{[0,1]}^3$, $(x,y+1,z)+{[0,1]}^3$, or $(x,y,z+1)+{[0,1]}^3$. In other words, edges of ${\mathscr{H}}_R$ go from a cube to one of its three face-adjacent cubes of height exactly one unit higher.

2. (b)

   The projection of this edge by $\phi$, $\phi (x,y,z)\to \phi (x^{\prime },y^{\prime },z^{\prime })$, is an edge of the triangular grid belonging to the region, that is, $\phi (e)\in {\mathrm{△}}_R^1$.

When the boundary of the region folds back onto itself, Condition (b) prevents the introduction of spurious edges. By construction, the edges of the ascendant graph ${\mathscr{H}}_R$ correspond to square faces of the cubic grid whose projections by $\phi$ are either entirely contained in $R$ or overlap boundary edges of $R$.

The ascendant graph ${\mathscr{H}}_R$ is acyclic, hence a directed acyclic graph (DAG). Moreover, it is invariant under translations by vectors of the form $(k,k,k)$ with $k\in \mathbb{Z}$, and therefore admits a $\mathbb{Z}$-periodic structure. This periodicity is essential and will be exploited later.

We now return to the roof associated with a lozenge tiling $T$ of a simply connected region $R$ and interpret it as a directed cut of the ascendant graph ${\mathscr{H}}_R$.

Each square face of the roof is adjacent to a pair of cubes $(x,y,z)+{[0,1]}^3$ and $(x^{\prime },y^{\prime },z^{\prime })+{[0,1]}^3$, where $(x^{\prime },y^{\prime },z^{\prime })$ equals one of $(x+1,y,z)$, $(x,y+1,z)$, or $(x,y,z+1)$. Since the projection of this square face by $\phi$ lies in the region $R$, both cubes belong to ${\mathrm{\square }}_R^3$ and therefore define an edge of ${\mathscr{H}}_R$. It follows that the square faces of the roof correspond exactly to a subset of edges of the ascendant graph. This subset has an important property.

We now introduce the notion of a directed cut (or dicut) of a directed graph. A dicut is a partition of the vertex set into two non empty subsets that we denote $\mathrm{bottom}$ and $\mathrm{top}$ such that all edges crossing the cut are directed from $\mathrm{bottom}$ to $\mathrm{top}$. The difference between a simple cut and a dicut is illustrated in Fig. 14.

One of the properties that we use in the following is that a directed path in a directed graph cannot be cut twice by a dicut.

The roof of the tiling $T$ being a monotone surface, it provides a partition of the cubes of ${\mathrm{\square }}_R$ which are above and below it. This partition is a dicut of ${\mathscr{H}}_R$.

At this stage, the correspondence is only one-way. While the roof of a tiling of $R$ always induces a directed cut of the ascendant graph ${\mathscr{H}}_R$, the converse is not true in general: there exist dicuts of ${\mathscr{H}}_R$ whose projections (the projections of the square faces separating the two sides of the cut) do not form a valid tiling of the region $R$, because some projected lozenges may overlap boundary edges of $R$ or violate interior constraints.

From the algorithmic perspective of solving an instance of Tiling$(R,X_1,X_2)$, the central question therefore becomes the following: under which conditions does the projection of a dicut of the ascendant graph ${\mathscr{H}}_R$ yield a valid tiling of the region $R$ satisfying the non overlapping and saliency constraints encoded by $X_1$ and $X_2$?

The first type of constraint enforces that no lozenge overlaps either a boundary edge of $\partial R$ or an interior edge belonging to $X=X_1\cup X_2$. Consider, for instance, an edge of the triangular grid between the vertices $\phi (x,y,z)$ and $\phi (x,y,z+1)$ that must not be overlapped by a lozenge. By Remark 7, the lozenge overlapping this edge is exactly the projection of the square face separating the cubes $(x,y,z)+{[0,1]}^3$ and $(x,y,z+1)+{[0,1]}^3$.

Therefore, forbidding this overlap is equivalent to requiring that the directed edge of ${\mathscr{H}}_R$ from $(x,y,z)+{[0,1]}^3$ to $(x,y,z+1)+{[0,1]}^3$ does not belong to the dicut. The same restriction applies to all translated pairs $(x+k,y+k,z+k)+{[0,1]}^3$ and $(x+k,y+k,z+k+1)+{[0,1]}^3$ for any $k\in \mathbb{Z}$. In other words, all directed edges between these pairs of cubes are declared unbreakable. Conversely, if a directed cut does not cut any of these unbreakable edges, then the corresponding lozenge overlapping the edge $\phi (x,y,z),\phi (x,y,z+1)$ cannot appear in the projection. Equivalent claims hold for the two other directions. It shows that by declaring unbreakable some pairs of cubes $(x+k,y+k,z+k)+{[0,1]}^3$ and $(x+k,y+k,z+k+1)+{[0,1]}^3$ for any $k\in \mathbb{Z}$, we exactly encode the non overlapping constraints.

We now consider the saliency constraints imposed by the edges of $X_2$. Assume that the edge $e$ between the vertices $\phi (x,y,z)$ and $\phi (x,y,z+1)$ belongs to $X_2$. The saliency constraint requires that the two lozenges adjacent to $e$ have distinct orientations.

To encode this condition in the ascendant graph, we introduce four families of cubes for each integer $k$:

as illustrated in Fig. 15. Each of these cubes has a face whose projection is a lozenge adjacent to or overlapping the edge $e$.

With this notation, the non overlapping constraint on $e$ is expressed by declaring the edge from $F_k$ to $B_{k+1}$ unbreakable. We now show how the saliency constraint itself can be enforced. The ascendant graph ${\mathscr{H}}_R$ contains the following chain of cubes: $\mathrm{\cdots }\to B_k\to L_k\text{ and }{R}_k\to F_k\to B_{k+1}\to \mathrm{\cdots }$ Any directed cut must intersect this infinite periodic chain at least once, otherwise one side of the cut would be empty. Several cutting patterns are possible, as shown in Fig. 16.

Cutting the two edges $B_k\to L_k$ and $B_k\to R_k$ satisfies the saliency constraint (Case 1 in Fig. 16). In contrast, cutting $B_k\to R_k$ together with $L_k\to F_k$ (Case 2), or symmetrically cutting $B_k\to L_k$ together with $R_k\to F_k$ (Case $2^{\prime }$), produces two adjacent lozenges of the same orientation and therefore violates the saliency constraint. Cutting both edges $L_k\to F_k$ and $R_k\to F_k$ is valid (Case 3). Finally, cutting $F_k\to B_{k+1}$ violates the non overlapping constraint on $e$.

The conclusion is that the saliency constraint on the edge $e$ is satisfied if and only if the two cubes $L_k$ and $R_k$ lie on the same side of the dicut. Equivalently, for every $k\in \mathbb{Z}$, the pair $L_k,R_k$ must not be separated by the dicut. Thus, the saliency constraint is enforced by declaring an unbreakable edge between $L_k$ and $R_k$ for all $k$.

A simple way to enforce that two vertices $u$ and $v$ of a directed graph belong to the same side of any dicut is to add a pair of opposite directed edges: one from $u$ to $v$ and one from $v$ to $u$. Indeed, no dicut can separate $u$ and $v$ without cutting at least one of these edges in a wrong direction.

This observation provides a simple mechanism for incorporating non overlapping and saliency constraints into the ascendant graph. We denote by ${\mathscr{H}}_R^{X_1,X_2}$ the enhanced ascendant graph, obtained from ${\mathscr{H}}_R$ by adding, for every unbreakable constraint induced by $\partial R$, $X_1$, and $X_2$, a pair of edges in opposite directions between the corresponding cubes.

We can now state the central correspondence between tilings and dicuts.

Moreover, the corresponding dicut is unique up to a global translation along the vector $\mathrm{𝟙}$, reflecting the classical height-shift ambiguity.

The proof of Theorem 9 is simply obtained by merging the previous arguments. The main idea is the translation of the non overlapping and saliency constraints into the unbreakable edges added in the enhanced ascendant graph ${\mathscr{H}}_R^{X_1,X_2}$. The detailed proof can be found in the full version available on ArXiv.

The goal of the next steps is to transform the problem of computing a dicut in the periodic graph ${\mathscr{H}}_R^{X_1,X_2}$ into a system of difference constraints. This type of reduction is not specific to the graph ${\mathscr{H}}_R^{X_1,X_2}$: it applies more generally to $\mathbb{Z}$-periodic directed graphs. Since a full general treatment is beyond the scope of this paper, we only illustrate the mechanism in Fig. 17, using $\mathbb{Z}$-periodic graphs different from ${\mathscr{H}}_R^{X_1,X_2}$. The purpose of this illustration is to provide intuition for why dicuts in $\mathbb{Z}$-periodic graphs can be characterized and computed via systems of difference constraints.

A classical result on systems of difference constraints states that such a system is feasible if and only if the associated weighted directed graph contains no cycle of strictly negative total weight. These two cases are illustrated in Fig. 11. It shows the next corollary of Theorem 3.

When there is no cycle of strictly negative weight, a feasible height function can be computed using classical shortest-path algorithms for graphs with possibly negative weights, such as the Bellman-Ford algorithm [1]. The resulting height function directly defines a dicut of ${\mathscr{H}}_R^{X_1,X_2}$ and therefore a valid tiling of the region $R$. It provides the first algorithm that we now discuss with its variants.

As in Thurston’s algorithm, we first examine the boundary of the region $R$. If the directed boundary cycles has nonzero total weight, then $R$ is not tilable and we stop. We therefore assume that the boundary cycle has total weight zero.

We temporarily ignore the interior constraints and work in the cube graph ${\mathscr{H}}_R^{\mathrm{\varnothing },\mathrm{\varnothing }}$. The boundary cycle of $R$ can be lifted to a directed cycle $c$ in this graph. By periodicity, this yields an infinite stack of directed cycles, each translated from the previous one by the vector $\mathrm{𝟙}=(1,1,1)$. If the region $R$ is tilable, the graph ${\mathscr{H}}_R^{\mathrm{\varnothing },\mathrm{\varnothing }}$ admits a dicut, and each cycle of this stack lies entirely either in the bottom or in the top part of the cut. Consequently, there exists a pair of consecutive cycles $c_0$ and $c_1$ such that $c_0$ lies in the bottom part and $c_1$ in the top part of the dicut. Then all cubes that can reach a cube of $c_0$ by a directed path belong to the lower part of a new dicut, which we call the lower cut. Dually, all cubes reachable from a cube of $c_1$ belong to the upper part of a second new dicut, called the upper cut. Any dicut of ${\mathscr{H}}_R^{X_1,X_2}$ separating $c_0$ and $c_1$ must lie between these two extremal cuts.

The first step of the algorithm therefore consists in computing the lower and upper dicuts of ${\mathscr{H}}_R^{\mathrm{\varnothing },\mathrm{\varnothing }}$ using Thurston’s algorithm. Either this step determines that $R$ is not tilable, or it provides the two extremal dicuts.

Assuming the first step succeeds, we now incorporate the interior constraints given by $X_1$ and $X_2$. These constraints introduce additional directed edges in the enhanced graph ${\mathscr{H}}_R^{X_1,X_2}$, which may create new predecessors for cubes in the lower cut.

The second step consists of a graph traversal starting from the lower cut: we iteratively add to the lower part of the lower dicut all cubes that have a directed edge leading to an already known cube of the lower part.

If during this process a cube belonging to the top part of the upper cut is added, then no dicut separating $c_0$ and $c_1$ exists. In this case, the instance Tiling$(R,X_1,X_2)$ is not feasible. Otherwise the growth of the bottom part of the lower dicut eventually ends when it remains no predecessor to add, providing a dicut of the graph ${\mathscr{H}}_R^{X_1,X_2}$ and thus a tiling solution of Tiling$(R,X_1,X_2)$ .

The time complexity of the first step is that of Thurston’s algorithm. The second step visits only cubes lying between the lower and upper dicuts of ${\mathscr{H}}_R^{\mathrm{\varnothing },\mathrm{\varnothing }}$. The vertical distance between these two cuts is bounded by the length of the boundary, namely $|\partial R|$. Therefore, the number of cubes in this intermediate region is at most $O(|R||\partial R|)$, which bounds the running time of the second step.

Overall, the advancing surface algorithm runs in time $O(|R||\partial R|)$.

In the particular case of the Calisson puzzle Tiling$({\mathrm{⬡}}_n,\mathrm{\varnothing },X_2)$, the lower and upper cuts are trivial: they correspond respectively to the empty and full $n\times n\times n$ cube. Since $|R|=\mathrm{\Theta }(n^2)$ and $|\partial R|=\mathrm{\Theta }(n)$, the advancing surface algorithm runs in time $O(n^3)$.

Although the mathematical framework underlying the advancing surface algorithm is nontrivial, its implementation in the context of the Calisson puzzle is remarkably simple. Using only a pencil, an eraser, and a three-dimensional intuition, one can simulate on the paper the successive additions of cubes. The previous analysis guarantees that this naive-looking procedure never misses a solution when one exists. Figure 18 illustrates several steps of the algorithm applied to the largest instance of Fig. 2.