Lozenge Tiling by Computing Distances

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.