Generation of Rosary permutations expressed in Hamiltonian circuits

Abstract

Systematic generation of a specific class of permutations fundamental to scheduling problems is described. In a nonoriented complete graph with n vertices, Hamiltonian circuits equivalent to 1/2( n - 1)! specific permutations of n elements, termed rosary permutations, can be defined. Each of them corresponds to two circular permutations which mirror-image each other, and is generated successively by a number system covering 3·4· ··· ·( n - 1) sets of edges. Every set of edges { e k }, 1 ≤ e k ≤ k , 3 ≤ k ≤ n - 1 is determined recursively by constructing a Hamiltonian circuit with k vertices from a Hamiltonian circuit with k - 1 vertices, starting with the Hamiltonian circuit of 3 vertices. The basic operation consists of transposition of a pair of adjacent vertices where the position of the pair in the permutation is determined by { e k }. Two algorithms treating the same example for five vertices are presented. It is very easy to derive all possible n ! permutations from the 1/2( n - 1)! rosary permutations by cycling the permutations and by taking them in the reverse order—procedures which can be performed fairly efficiently by computer.