Cayley Graph Techniques for Permutation Routing on Bus Interconnection Networks

Cayley graphs and elementary group theory are used to eeciently nd minimum length permutation routes in a bus interconnection network. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying point-to-point routing 1, 2, 3, 4], but the technique has not been widely used for the more important problem of permutation routing. This is due to the potentially explosive growth in both the size of the graph and the number of generating permutations, referred to as one-step permutation routes, used to deene the underlying graph. This paper describes a method for moderating that growth, using techniques from 6, 7], and applies that method to a model of bus interconnection networks. Some techniques from 6] are rst reviewed. They show how the existence of a natural set of graph automorphisms that can be used to construct a homomorphism into a much smaller, reduced multigraph, from which shortest permutation routes can be recovered. In a particularly striking example, a fully connected network of 16 chips and 4 bus lines, involving 1:0 10 17 permutations (nodes of the Cayley graph), is reduced to a computation on a graph with only 3;950 nodes. A new group-theoretic characterization of the redundant edges of that multigraph is then presented, and experimental results are given. The improved computation time is roughly proportional both to the factor of reduction in the number of nodes and the factor of reduction in the number of edges. Generalizations to other interconnection networks are also discussed.