Counting polyominoes: a parallel implementation for cluster computing
ICCS'03 Proceedings of the 2003 international conference on Computational science: PartIII
Counting d-dimensional polycubes and nonrectangular planar polyominoes
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
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A d-D polycube of size nis a connected set of ncells (hypercubes) of an orthogonal d-dimensional lattice, where connectivity is through (d茂戮驴 1)-dimensional faces of the cells. Computing Ad(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) polynomialin both nand d. Furthermore, we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously. This enables us to find Ad(n) for values of dand nfar beyond any previous attempt.