Lozenge Tiling by Computing Distances

2026-03-02

AI Summary

This study addresses the Calisson tiling problem: tiling a hexagonal region on a triangular lattice with rhombi of three orientations, such that specified edges serve as tile boundaries and adjacent rhombi have distinct orientations. The authors propose a polynomial-time algorithm—termed the advancing surface algorithm—with cubic time complexity, which reformulates the tiling problem as constructing a height function corresponding to a three-dimensional monotone stepped surface. The solution leverages directed cuts and a difference constraints system. This work offers the first reinterpretation of Conway and Thurston’s classical theory from a theoretical computer science perspective, introducing a novel framework based on periodic directed graphs and difference constraints. It yields the first efficient algorithm for deciding tileability of finite regions and extends to infinite grids with local constraints even in the absence of global boundary conditions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Lozenge Tiling
• Calisson Puzzle
• Tiling Constraints
• Triangular Grid
• Monotone Stepped Surfaces

Innovation

• lozenge tiling
• difference constraints
• directed cuts
• height functions
• polynomial-time algorithm