A theory for total exchange in multidimensional interconnection networks

Total exchange (or multiscattering) is one of the important collective communication problems in multiprocessor interconnection networks. It involves the dissemination of distinct messages from every node to every other node. We present a novel theory for solving the problem in any multidimensional (cartesian product) network. These networks have been adopted as cost-effective interconnection structures for distributed-memory multiprocessors. We construct a general algorithm for single-port networks and provide conditions under which it behaves optimally. It is seen that many of the popular topologies, including hypercubes, k-ary n-cubes, and general tori satisfy these conditions. The algorithm is also extended to homogeneous networks with 2^k dimensions and with multiport capabilities. Optimality conditions are also given for this model. In the course of our analysis, we also derive a formula for the average distance of nodes in multidimensional networks; it can be used to obtain almost closed-form results for many interesting networks