Abstract
The average internode distancein an interconnection network (or its average distance for short) is an indicator of expected message latency in that network under light and moderate network traffic. Unfortunately, it is not always easy to find an exact value for the average internode distance, particularly for networks that are not node-symmetric, because the computation must be repeated for many classes of nodes. In this short paper, we derive exact formulas for the average internode distance in mesh and complete binary tree networks.
Highlights
Introduction interconnection network, exact formulas for it have not been published for many useful networks, including the two widely used ones: meshes and binary trees
A variety of interconnection networks have been studied for linking the nodes in a parallel or distributed system [1,2,3,4]
We focus on a particular static attribute of a network, its average internode distance ∆, and derive exact formulas for it in the case of two highly popular, but node-asymmetric, networks
Summary
×nq mesh are labeled by q-tuples x1x2...xq, wherexi (1 ≤xi≤ni) is the dimension-i address of the node. A q-dimensional torus is defined, except that every node has exactly 2q neighbors due to the inclusion of wraparound links that connect the last node along each dimension to the first node. The shortest path length in a q-dimensional mesh can be found by adding the distances of the destination node from the source node along each of the q dimensions. The average internode distance in a q-dimensional n1×n2× ... When the dimensions ni are large, the average internode distance in (3) is roughly one-third of the diameter in (4): DqD-mesh = Σ1≤i≤qni – q (4). Note that the average internode distance of a p-ring is p/4 when p is even and it is slightly less when p is odd. Note that the ∆/D ratio for a ring or torus is always 1/2
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