A necessary and sufficient closure property for two-stage constructions of switching networks

Two-stage constructions and banyan-type networks play important roles in designing high speed switch fabrics. Switch fabrics designed from two-stage constructions and banyan-type networks cannot realize all the permutations and are known as conditionally nonblocking switches. A renowned property for conditionally nonblocking switches is the closure property, i.e., if all the switches in a two-stage construction can realize a subset of permutations that satisfy a certain property P, then the switch resulted from the two-stage construction can also realize a subset of permutations that satisfy the same property P. However, such a closure property is mostly stated for the sufficient part in the literature and the necessary part of the statement is in general either not true or unknown. Finding such a necessary and sufficient result is of fundamental importance to two-stage constructions as it can completely characterize the permutations that are realizable by two-stage constructions. In this paper, we prove a necessary and sufficient closure property for two-stage constructions. For this, we consider uniform mapping permutations and define uniform mapping switches as switches that can only realize the set of uniform mapping permutations. We show that a switch constructed by a two-stage construction is a uniform mapping switch if and only if all the switches at the first stage and the second stage are uniform mapping switches. Such a necessary and sufficient result provides a complete characterization of the permutations that can be realized by two-stage constructions. Since banyan-type networks are constructed recursively by using two-stage constructions with 2 × 2 switches, we obtain a complete characterization for any realizable permutation of any banyan-type network as a composition of its trace, a uniform mapping permutation and its guide.