Parallel Enumeration of Lattice Animals

Abstract

Lattice animals are connected sets of lattice cells. When the lattice is in d dimensions, connectedness is through (d − 1)-dimensional features of the lattice. For example, connectedness of two-dimensional animals (e.g., on the rectangular, triangular, and hexagonal lattices) are through edges, connectedness of 3-dimensional polycubes is through faces, etc. Much attention has been given in the literature to algorithms for counting animals of a given size (number of cells) on different lattices. One such algorithm was suggested in 1981 by Redelmeier for counting polyominoes (animals on the 2D orthogonal lattice). This was the first algorithm that generated polyominoes without repetitions. In previous works we extended this algorithm to other lattices and showed how to avoid its (originally) huge memory consumption. In the current paper we describe how to parallelize the extended algorithm. Our implementation runs on the Internet, effectively using an unlimited number of computers running portions of the computation. Thus, we were able to extend the known counts of animals on many types of lattices with values which were previously out of reach.