Using semidirect products of groups to build classes of interconnection networks

Abstract

We build a framework within which we can define a wide range of Cayley graphs of semidirect products of abelian groups, suitable for use as interconnection networks and which we call toroidal semidirect product graphs. Our framework encompasses various existing interconnection networks such as cube-connected cycles, recursive cubes of rings, cube-connected circulants and dual-cubes, as well as certain multiswapped networks, pruned tori and biswapped networks; it also enables the construction of new hitherto uninvestigated but highly structured interconnection networks. We go on to design an efficient shortest-path routing algorithm that can be applied to any graph that can be defined within our framework. Our algorithm runs in time that is polylogarithmic in the size of the base group and polynomial in the size of the extending group of the given semidirect product. We also obtain analytic upper bounds on the diameters of our toroidal semidirect product graphs.