Dense, symmetric interconnection networks

Widespread use of computer systems with thousands of processors, or processing elements, is still in the future even though the mesh-connected Massively Parallel Processor (MPP) and the hypercube-connected Connection Machine are already in this category. To achieve efficient massively parallel systems, there is a continuing search for dense and symmetric interconnection graphs. A dense and symmetric network allows identical message routing schemes at every vertex and reduces communication delay. Although symmetric, neither the (toroidal) mesh nor the hypercube are very dense. The densest, regular graphs have randomly interconnected vertices, but they are non-constructive and obviously not symmetric. However, certain group-theoretic graphs, Cayley Graphs, are known to be symmetric and some of the densest graphs have been identified as Cayley Graphs based on the Borel subgroups, or the Borel Cayley graphs. The objective of our research is to study the application of these Borel Cayley graphs as interconnection networks. Our contributions encompass three issues: (i) systematic representation and layout; (ii) time and space efficient routing algorithms and related network performance in the presence of contention; (iii) theoretical results, including relationships among various parameters of Borel Cayley graphs and resultant formulations in one and two dimensional integer domains. These last results lead to a new graph classification: symmetric, pseudo-random graphs. We proposed the novel concept of using the well-defined, integer labeled, Generalized Chordal Rings (GCR) and Chordal Rings (CR) as representations of Cayley graphs. This concept solved the problem of the lack of a systematic representation and layout of the original Cayley graphs in their application as interconnection networks. Once Cayley graphs are represented in the integer domain of GCR/CR, we exploit their properties to devise several routing algorithms. The algorithms apply to Cayley graphs in general and Borel Cayley graphs in particular and have different time and space measures. The network performance was studied by simulation and a probabilistic model. An interesting conclusion from the simulation results is that a simple, oblivious, bufferless algorithm works well on Borel Cayley graphs because the pseudo-random structure naturally produces uniform traffic distribution. Finally, we studied the relationships of various parameters of a Borel Cayley graph. Based on the resultant findings, new formulations in two-dimensional and one-dimensional integer domains were defined. An important implication of such formulations is the simplification of Borel Cayley graphs to symmetric, pseudo-random graphs.