An Optimal Algorithm for Tiling the Plane with a Translated Polyomino

TL;DR

This paper presents an efficient linear-time algorithm to determine and enumerate regular tilings of the plane using translated polyominoes, establishing bounds on the number of such tilings.

Contribution

It introduces a novel $O(n)$-time algorithm for tiling determination and enumeration, along with a proof of the maximum number of regular tilings.

Findings

Linear-time algorithm for tiling determination

Enumeration of all regular tilings in $O(n)$ time

Proof that at most $ heta(n)$ such tilings exist

Abstract

We give a $O(n)$-time algorithm for determining whether translations of a polyomino with $n$ edges can tile the plane. The algorithm is also a $O(n)$-time algorithm for enumerating all such tilings that are also regular, and we prove that at most $\Theta(n)$ such tilings exist.