Randomized routing of virtual connections in essentially nonblocking log N-depth networks

An optimal N×N circuit switching network with θ(N×N) bandwidth has a lower bound of θ(N·log N) hardware, which includes all crosspoints, bits of memory and logic gates, and a lower bound of θ(logN) set-up time. To date no known self-routing circuit switching networks with explicit constructions achieve these lower bounds. The authors consider a randomized routing algorithm on a class of circuit switching networks called “extended dilated banyans”. It is proven that the blocking probability of an individual connection request is O[log_b N·(k/d)^d], where d is the dilation factor and k is a constant. With a dilation of θ(log log N) and a loading <1 the blocking probability is shown to approach zero, yielding an “essentially nonblocking” network. The hardware complexity of these networks depends upon the internal node implementation. A space division node yields a network with θ(N·log N·log log N) hardware and θ(log N·log log N) set-up time. A time division node, in which the bits from each connection are dynamically concentrated in time using a “time-bit-concentrator” circuit, yields a network with an asymptotically optimal O(N·log N) hardware and a slightly suboptimal θ(log N·log log N) set-up time. Both implementations improve upon the best known explicit constructions of self-routing circuit switching networks with θ(N) bandwidth, and the TDM construction meets Shannon´s lower bound on the cost of such networks. It is shown that extended dilated banyans can carry significantly more traffic than the Batcher-banyan switch and its variants