Performance, algorithmic, and robustness attributes of perfect difference networks

Perfect difference networks (PDNs) that are based on the mathematical notion of perfect difference sets have been shown to comprise an asymptotically optimal method for connecting a number of nodes into a network with diameter 2. Justifications for, and mathematical underpinning of, PDNs appear in a companion paper. In this paper, we compare PDNs and some of their derivatives to interconnection networks with similar cost/performance, including certain generalized hypercubes and their hierarchical variants. Additionally, we discuss point-to-point and collective communication algorithms and derive a general emulation result that relates the performance of PDNs to that of complete networks as ideal benchmarks. We show that PDNs are quite robust, both with regard to node and link failures that can be tolerated and in terms of blandness (not having weak spots). In particular, we prove that the fault diameter of PDNs is no greater than 4. Finally, we study the complexity and scalability aspects of these networks, concluding that PDNs and their derivatives allow the construction of very low diameter networks close to any arbitrary desired size and that, in many respects, PDNs offer optimal performance and fault tolerance relative to their complexity or implementation cost.