On the sizes of permutation networks and consequences for efficient simulation of hypercube algorithms on bounded-degree networks

Abstract

The sizes of permutation networks for special sets of permutations are investigated. The study of the planar realization and the search for small but hard sets of permutations are also included. Several asymptotically optimal estimations for distinct subsets of the set of all permutations are established here. The two main results are: (i) an asymptotically optimal permutation network of size 6.N.log log N for shifts of power 2. (ii) an asymptotically optimal planar permutation network of size (Θ(N2. (log log N/log N)2) for shifts of power 2. A consequence of our results is a construction of a 4-degree network which can simulate each communication step of any hypercube algorithm using edges from at most a constant number of different dimensions in one step in O(log log N) communication steps. A new sorting network as well as an essential improvement of gossiping in vertex-disjoint path mode in bounded-degree networks follow.