A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

TL;DR

This paper presents an efficient algorithm with quasilinear time complexity for determining whether a given polyomino can tile the plane isohedrally, improving previous computational methods significantly.

Contribution

The authors develop a $O(n\log^2 n)$-time algorithm for the isohedral tiling decision problem, advancing computational geometry techniques for polyomino tilings.

Findings

The new algorithm runs in quasilinear time, significantly faster than prior methods.

It successfully decides isohedral tiling possibilities for polyominoes with n edges.

The approach generalizes previous work and improves computational efficiency.

Abstract

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a $O(n\log^2{n})$-time algorithm for deciding if a polyomino with $n$ edges can tile the plane isohedrally. This improves on the $O(n^{18})$-time algorithm of Keating and Vince and generalizes recent work by Brlek, Proven\c{c}al, F\'{e}dou, and the second author.