An algorithm for deciding if a polyomino tiles the plane by translations

An algorithm for deciding if a polyomino tiles the plane

For polyominoes coded by their boundary word, we describe a quadratic O(n) algorithm in the boundary length n which improves the naive O(n) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.

On the problem of deciding if a polyomino tiles the plane by translation

An Optimal Algorithm for Tiling the Plane with a Translated Polyomino

We give aO(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 regular tilings, and we prove that at most Θ(n) such tilings exist.

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

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 log2 n)-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provença...

Tiling the Plane with a Fixed Number of Polyominoes

Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the problem is undecidable for sets of 11 polyominoes.