Zone-Balanced Networks and Block Designs

We shall consider multistage switching networks composed of rectangular switches. For an input terminal u and an output terminal u', the channel graph for u and u', denoted by G(u,u'), is defined to be the union of all paths that can be used to connect u and u'. (A channel graph is also called a linear graph.) A network is said to be balanced if all channel graphs G(u,u'), where u is in the set I of input terminals and u' is in the set Q of output terminals, are isomorphic.1,2 A network is said to be zone-balanced if it has two nonisomorphic channel graphs, say Gi and G2, so that the channel graph G{u,u') is isomorphic to G if u and u' are in the same zone, and G{u,u') is isomorphic to G2 if u and u' are in different zones. (See Fig. 1. Note that the switches in the network can be viewed as nodes of the corresponding graph.) G is called the internal graph and G2 is called the external graph of the switching network. A zone-balanced network usually consists of three parts. The primary part consists of a few stages where traffic distribution takes place within each zone so that each input terminal has sufficient access to the central stage. The secondary part is the central stage which provides interconnections between different zones. The tertiary part plays the same role for the output terminals as the primary part does for the input terminals. 2957