Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino

Abstract

Given N distinct memory modules, the elements of an (infinite) array in storage are distributed 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 Shapiro ( IEEE Trans. Comput. C-27 (1978), 421–428) proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. A conjecture of Shapiro is settled and it is proved 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.