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What Designers of Bus and Network Architectures Should Know about Hypercubes
April 2003 (vol. 52 no. 4)
pp. 525-544

Abstract—We quantify why, as designers, we should prefer clique-based hypercubes (K-cubes) over traditional hypercubes based on cycles (C-cubes). Reaping fresh analytic results, we find that K-cubes minimize the wirecount and, simultaneously, the latency of hypercube architectures that tolerate failure of any f nodes. Refining the graph model of Hayes (1976), we pose the feasibility of configuration as a problem in multivariate optimization:

What (f + 1){\hbox{-}}{\rm connected}n{\hbox{-}}{\rm vertex} graphs with fewest edges \lceil n ( f + 1) / 2\rceil minimize the maximum a) radius or b) diameter of subgraphs (i.e., quorums) induced by deleting up to f vertices? (1)

We solve (1) for f that is superlogarithmic but sublinear in n and, in the process, prove: 1) the fault tolerance of K-cubes is proportionally greater than that of C-cubes; 2) quorums formed from K-cubes have a diameter that is asymptotically convergent to the Moore Bound on radius; 3) under any conditions of scaling, by contrast, C-cubes diverge from the Moore Bound. Thus, K-cubes are optimal, while C-cubes are suboptimal. Our exposition furthermore: 4) counterexamples, corrects, and generalizes a mistaken claim by Armstrong and Gray (1981) concerning binary cubes; 5) proves that K-cubes and certain of their quorums are the only graphs which can be labeled such that the edge distance between any two vertices equals the Hamming distance between their labels; and 6) extends our results to K-cube-connected cycles and edges. We illustrate and motivate our work with applications to the synthesis of multicomputer architectures for deep space missions.

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Index Terms:
Hypercube fault tolerance, hypercube latency, configuration architectures, performability, quorums, Hamming graphs, K-cubes, Moore graphs, Moore Bound, C-cubes, Lee distance.
Citation:
Laurence E. LaForge, Kirk F. Korver, M. Sami Fadali, "What Designers of Bus and Network Architectures Should Know about Hypercubes," IEEE Transactions on Computers, vol. 52, no. 4, pp. 525-544, April 2003, doi:10.1109/TC.2003.1190592
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