Products of Networks with Logarithmic Diameter and Fixed Degree

This paper rst presents some general properties of product networks pertinent to parallel ar-chitectures and then focuses on three case studies. These are products of complete binary trees, shuue-exchange, and de Bruijn networks. It is shown that all of these are powerful architectures for parallel computation, as evidenced by their ability to eeciently emulate numerous other archi-tectures. In particular, r-dimensional grids, and r-dimensional meshes of trees can be embedded eeciently in products of these graphs, i.e. either as a subgraph or with small constant dilation and congestion. In addition, the shuue-exchange network can be embedded in r-dimensional product of shuue exchange networks with dilation cost 2r and congestion cost 2. Similarly, the de Bruijn network can be embedded in r-dimensional product of de Bruijn networks with dilation cost r and congestion cost 4. Moreover, it is well known that shuue-exchange and de Bruijn graphs can emulate the hypercube with a small constant slowdown for \normal" algorithms. This means that their product versions can also emulate these hypercube algorithms with constant slowdown. Conclusions include a discussion of many open research areas.