Programming Cellular Permutation Networks Through Decomposition of Symmetric Groups

A fundamental problem in interconnection network theory is to design permutation networks with as few cells as possible and a small programming or setup time. The well-known networks of Benes and Waksman have asymptotically optimal cell counts, but the best setup algorithm available for such networks with n inputs requires O(n log2 n) sequential time. As an alternative, this paper considers another class of permutation networks which are collectively referred to as cellular permutation arrays. Using a group theoretic formulation, a natural correspondence is established between such permutation networks and iterative decompositions of symmetric groups through cosets. Based on this correspondence, the setup problem is reduced to iteratively determining the leaders of the cosets to which the permutation to be realized belongs. This, in turn, leads to linear-time setup algorithms for cellular permutation arrays. The paper describes these algorithms in detail for two families of cellular permutation arrays reported in the literature.