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

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2 Author(s)
T. Szymanski ; Dept. of Electr. Eng., McGill Univ., Montreal, Que., Canada ; Chien Fang

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[logb 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

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IEEE Transactions on Communications  (Volume:43 ,  Issue: 9 )