Optimal realization of any BPC permutation on K-extra-stage Omega networks

An N×N k-Omega network is obtained by adding k more stages in front of an Omega network. An N-permutation defines a bijection between the set of N sources and the set of N destinations. Such a permutation is said to be admissible to a k-Omega if N conflict-free paths, one for each source-destination pair defined by the permutation, can be established simultaneously. When an N-permutation is not admissible, it is desirable to divide the N pairs into a minimum number of groups (passes) such that the conflict-free paths can be established for the pairs id each group. Raghavendra and Varma solved this problem for BPC (Bit Permutation Complement) permutations on an Omega without extra stage. This paper generalizes their result to a k-Omega where k can be any integer between 0 and n-1. An O(NlgN) algorithm is given which realizes any BPC permutation in a minimum number of passes on a k-Omega (0⩽k⩽n-1)