Periodic versus arbitrary tessellations of the plane using polyominos of a single type

Abstract

Given N parallel memory modules, we like to distribute the elements of an (infinite) array in storage such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector-processing and SIMD machines. In 1975 H.D. Shapiro proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. We settle a conjecture of Shapiro and prove that for polyominos P a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme. (Periodicity implies a rapid technique to locate data elements.) The proof shows that when a polyomino P tessellates the plane without rotations or reflections, then it can tessellate the plane periodically, i.e., with the instances of P arranged in a lattice. It is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane, assuming the polyominos in the tessellation should all have an equal orientation.