New Methods for the Generation of Permutations, Combinations, and Other Combinatorial Objects in Parallel

Abstract

In this paper, three methods for generating sets of combinatorial objects in parallel are developed. The first one is generation of the next choice function satisfying a requirement W of a given indexed family. Then it is shown that permutations and combinations can be represented by these choice functions. The second method is decomposition of a family of indexed sets into layers, carried out in order to split the set of all possible choice functions of a given family into subsets. On this base there are developed parallel algorithms for generating regular and irregular sets of permutations or combinations. The third method is a fast algorithm for finding any kth combination of m out of n items. Theorem 5 proves the correctness of this algorithm and reveals the inner structure of a full set of combinations. Methods for the generation of full sets of combinations or certain "regular" sets of permutations are developed for the SIMD model of parallel processing. These are based on the enumeration of combinations.