Counting polycubes without the dimensionality curse

Abstract

A d-D polycube of size n is a connected set of n cells (hypercubes) of an orthogonal d-dimensional lattice, where connectivity is through (d - 1)-dimensional faces of the cells. Computing A d (n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present a method whose order of memory consumption is a (very low) polynomial in both n and d. Furthermore, we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously. This enables us to find A d (n) for values of d and n far beyond any previous attempt. © 2008 Springer-Verlag Berlin Heidelberg.