Optimal Rearrangeable Multistage Connecting Networks

A rearrangeable connecting network is one whose permitted states realize every assignment of inlets to outlets — that is, one in which it is possible to rearrange existing calls so as to put in any new call. In the effort to provide adequate telephone service with efficient networks it is of interest to be able to select rearrangeable networks (from suitable classes) having a minimum number of crosspoints. This problem is fully resolved for the class of connecting networks built of stages of identical square switches arranged symmetrically around a center stage: roughly, the optimal network should have as many stages as possible, with switches that are as small as possible, the largest switches being in the center stage; the cost (in crosspoints per inlet) of an optimal network of N inlets and N outlets is nearly twice the sum of the prime divisors of N, while the number of its stages is 2×−1, where x is the number of prime divisors of N, in each case counted according to their multiplicity. By using a large number of stages, these designs achieve a far greater combinatorial efficiency than has been attained heretofore.