GDE on Tori and Meshes

Abstract

In the last chapter, we considered the hypercube and some other structures and their optimal pairwise averaging operation in GDE load balancing. One important finding is that the product of any two such structures also has this property. In this chapter, we consider the structures of n-dimensional torus and mesh (n ≥ 1), and their special cases, the ring, the chain, and the k-ary n-cube. An n-dimensional torus is a variant of the mesh with end-round connections, as illustrated in Figure 4.1 (the end-round connections at the back and bottom are omitted in the figure for clarity). A k-ary n-cube is a network with n dimensions having k nodes in each dimensions [4, 145]. It is a special case of the n-dimensional torus which allows different orders in different dimensions. The order of a dimension refers to the number of nodes in the dimension. The hypercube is a special case of both the n-dimensional mesh and the k-ary n-cube. A hypercube is an n-dimensional mesh with the same order of 2 in each dimension, that is, a 2-ary n-cube. We limit our scope to these structures because they are among the most popular choices of topologies for today’s parallel computers [145, 163, 189].