Simulating parallel neighboring communications among square meshes and square toruses

Abstract

Given a d-dimensional square mesh or square torus G and a c-dimensional square mesh or square torus H such that G and H are of the same size but may differ in dimensions and shapes, we study the problem of simulating in H parallel neighboring communications in G. We assume that the nodes in H have only unit-size buffers associated with the links, and that packets can be sent and received simultaneously from all outbound links and inbound links of the nodes. For permutation-type parallel neighboring communications, for all the combinations of graph types and graph shapes of G and H, except for the case in which d < c and c is not divisible by d, we show that H can simulate G either optimally or optimally up to a constant multiplicative factor for fixed values of d and c. For scatter-type parallel neighboring communications, for some special cases of G and H, we also show that H can optimally simulate G. All these simulation times are smaller than the diameter of H, the lower bound on the routing complexity to support general data permutations in H.