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ÐHoneycomb and diamond networks have been proposed as alternatives to mesh and torus architectures for parallel processing. When wraparound links are included in honeycomb and diamond networks, the resulting structures can be viewed as having been derived via a systematic pruning scheme applied to the links of 2D and 3D tori, respectively. The removal of links, which is performed along a diagonal pruning direction, preserves the network's node-symmetry and diameter, while reducing its implementation complexity and VLSI layout area. In this paper, we prove that honeycomb and diamond networks are special subgraphs of complete 2D and 3D tori, respectively, and show this viewpoint to hold important implications for their physical layouts and routing schemes. Because pruning reduces the node degree without increasing the network diameter, the pruned networks have an advantage when the degree-diameter product is used as a figure of merit. Additionally, if the reduced node degree is used as an opportunity to increase the link bandwidths to equalize the costs of pruned and unpruned networks, a gain in communication performance may result.

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AbstractÐChordal rings have been proposed in the past as networks that combine the simple routing framework of rings with the lower diameter, wider bisection, and higher resilience of other architectures. Virtually all proposed chordal ring networks are nodesymmetric, i.e., all nodes have the same in/out degree and interconnection pattern. Unfortunately, such regular chordal rings are not scalable. In this paper, periodically regular chordal (PRC) ring networks are proposed as a compromise for combining low node degree with small diameter. By varying the PRC ring parameters, one can obtain architectures with significantly different characteristics (e.g., from linear to logarithmic diameter), while maintaining an elegant framework for computation and communication. In particular, a very simple and efficient routing algorithm works for the entire spectrum of PRC rings thus obtained. This flexibility has important implications for key system attributes such as architectural scalability, software portability, and fault tolerance. Our discussion is centered on unidirectional PRC rings with in/out-degree of 2. We explore the basic structure, topological properties, optimization of parameters, VLSI layout, and scalability of such networks, develop packet and wormhole routing algorithms for them, and briefly compare them to competing fixed-degree architectures such as symmetric chordal rings, meshes, tori, and cube-connected cycles.

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Incomplete or pruned k-ary n-cube, nX3; is derived as follows. All links of dimension n À 1 are left in place and links of the remaining n À 1 dimensions are removed, except for one, which is chosen periodically from the remaining dimensions along the intact dimension n À 1: This leads to a node degree of 4 instead of the original 2n and results in regular networks that are Cayley graphs, provided that n À 1 divides k: For n ¼ 3 ðn ¼ 5Þ; the preceding restriction is not problematic, as it only requires that k be even (a multiple of 4). In other cases, changes to the basis network to be pruned, or to the pruning algorithm, can mitigate the problem. Incomplete k-ary n-cube maintains a number of desirable topological properties of its unpruned counterpart despite having fewer links. It is maximally connected, has diameter and fault diameter very close to those of k-ary n-cube, and an average internode distance that is only slightly greater. Hence, the cost/performance tradeoffs offered by our pruning scheme can in fact lead to useful, and practically realizable, parallel architectures. We study pruned k-ary n-cubes in general and offer some additional results for the special case n ¼ 3: r

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