Optimal Constructions of Fault-Tolerant Multistage Interconnection Networks

In this paper we discover the family of Fault-Tolerant Multistage Interconnection Networks (MINs) that tolerates switch faults with a minimal number of redundant switching stages. While previously known constructions handled switch faults by eliminating complete stages, our approach is to bypass faulty switches by utilizing redundant paths. As a result, we are able to construct the first known fault-tolerant MINs that are optimal in the number of redundant stages. Our fault model assumes that a faulty switch can be bypassed and our goal is to guarantee arbitrary point to point and broadcast connectivity. Under this model, we show that to tolerate f switch faults the MIN must have at least f redundant stages. We then present the explicit construction of a MIN that meets this lower-bound. This construction repeatedly uses the singleton basis of the n-dimensional vector space as the mask vectors of the MIN. We generalize this construction and prove that an n-dimensional MIN is optimally fault-tolerant if and only if the mask vectors of every n consecutive stages span the n-dimensional vector space.