The complexity of routing in Clos permutation networks

A lower bound on the amount of information necessary to compute the switch settings for a three-stage Clos network is derived. The bound is derived by considering the effect of a family of permutations, called balanced multiloops, on the settings of the switches of a Clos network. By carefully selecting these permutations, it is proven that there exists at least one switch in each stage whose setting depends upon at least (k-3)(m/2+1)/2 assignments in the permutations to be routed for k⩾4 where m is the number of center-stage switches and k is the number of input and output-stage switches. It is also proven that the setting of the input stage depends on at least m-1 assignments. Lower bounds on the routing time of a three-stage Clos network on a variety of machine models follow immediately from these results. In particular, any constant fan-in implementation of any routing algorithm for such a network should have a time lower bound of Ω(log mk)