A linear algorithm for Brick Wang tiling

TL;DR

This paper introduces a linear algorithm for solving tiling problems with Brick Wang tiles, enabling efficient generation of wall patterns in computer graphics for arbitrary planar regions with holes.

Contribution

It generalizes previous results by providing a linear-time algorithm for tiling arbitrary planar regions with Brick Wang tiles, including regions with holes.

Findings

The algorithm efficiently decides tileability of complex regions.

It extends tiling solutions to regions with holes.

The method is applicable in computer graphics for pattern generation.

Abstract

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.