Counting Polycubes without the Dimensionality Curse
A d-D polycube of size n is a connected set of n cells (hy-percubes) of an orthogonal d-dimensional lattice, where connectivity is through (d-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)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 Ad(n) for values of d and n far beyond any previous attempt.
Gadi Aleksandrowicz Gill Barequet
Center for Graphics and Geometric Computing Dept. of Computer Science Technion-Israel Institute of Technology Haifa 32000, Israel
国际会议
The 4th Annual International Computing and Combinatorics Conference,COCOON 2008(第14届国际计算和组合会议)
大连
英文
100-109
2008-06-01（万方平台首次上网日期，不代表论文的发表时间）