Necessary Conditions and Empirical Observations for Rearrangeable Banyan-Type Networks

Abstract

A banyan-type network is constructed by aligning unit switches with two inlets and outlets in multiple stages. Rearrangeable banyan-type networks are crucial for applications such as communication systems because they can universally establish connections for any request without blocking. If the number of network inputs (or outputs) is 2^n (n > 0), the banyan-type network should have 2n − 1 or more stages to be rearrangeable. A few rearrangeable 2n − 1 stage networks have been reported. However, the class of rearrangeable 2n − 1 stage banyan-type networks has not been completely clarified. This study examines the identification of rearrangeable 2n − 1 stage banyan-type networks that are not isomorphic to one another. This is done by generating candidate networks and checking their rearrangeability via the satisfiability problem. The drawback of this approach is its poor scalability due to numerous candidates. To eliminate this drawback, it is shown that the candidates can be reduced to a smaller number of networks called pure banyan networks. This is achieved by analyzing network isomorphism. Next, necessary conditions are derived for rearrangeability. Utilizing the conditions, the number of candidate networks further decreases because blocking networks are identified and removed from the candidates. For the reduced number of candidate networks, rearrangeability is assessed through computer experiments for n = 4 and 5. For n = 4, the result shows that any rearrangeable configuration is isomorphic to previously reported rearrangeable networks. For n = 5, the blocking probability is extremely low and the rearrangeability is inconclusive for two groups of networks.