Periodically Regular Chordal Rings are Preferable to Double-Ring Networks

The susceptibility of ring networks to disconnection as a result of one or two node/link failures has led to a number of proposals to increase the robustness of such networks. One class of proposals, discussed primarily in the networking and communication communities, but also advocated for use in parallel and distributed systems, involves the provision of a second ring to improve system throughput during nonnal operation and to make alternate paths available in the event of node or link failures. With regard to the advantages just listed, chordal rings are quite similar to double-ring networks, and periodically regular chordal (PRC) rings offer the added benefit of smaller node degree compared with node-symmetric chordal rings of comparable diameters. In this paper, we note that certain double-ring networks are isomorphic to suitably constructed PRC rings, while other varieties correspond to PRC rings that closely approximate their static and dynamic attributes. These results, combined with greater flexibility and other advantages for the PRC-ring family of networks, demonstrate that PRC rings are preferable to double-ring networks in virtually all application contexts. A byproduct of our observations on the relationships among double-ring networks, generalized Petersen graphs, and PRC rings is that by amalgamating known results for these network classes, many more tools and techniques become applicable to the analysis and synthesis of robust ring networks for parallel and distributed computing. As examples of new results that can be developed with this viewpoint, we present near-optimal and fault-tolerant routing algorithms for our PRC ring networks.