Permutation routing via Cayley graphs with an example for bus interconnection networks

Abstract

Abstract. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying point-to-point routing [1,2,3,5,6,7,12]. The extension of these techniques to the more important problem of permutation routing on interconnection networks presents fundamental problems. 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 define the underlying graph. This paper describes a technique for moderating that growth so that the techniques in [8] can be applied for finding optimal permutation routes. In a particularly striking example, a bus interconnection architecture involving 1.0 × 10 17 permutations (nodes of the Cayley graph) is reduced to a computation on a graph with only 3,950 nodes. Further, it is shown how many of the 58,624 generators (directed edges labelled by one-step permutation routes) at each node of the graph may be eliminated as locally redundant.