On the Size of Permutation Networks and Consequences for Efficient Simulation of Hypercube Algorithms on Bounded-Degree Networks
Résumé
The sizes of permutation networks and planar permutation networks for special sets of permutations are investigated. 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 switching network of size O(N log log N) for shifts of power 2. (ii) an asymptotically optimal planar permutation network of size Θ(N 2 • (log log N/ log N) 2) for shifts of power 2. A consequence of our results is the 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 communication step in O(log log N) communication steps. An essential improvement of gossiping in vertex-disjoint path mode in bounded-degree networks follows.
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