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<p><b>Abstract</b>—We quantify why, as designers, we should prefer clique-based hypercubes (<it>K-cubes</it>) over traditional hypercubes based on cycles (<it>C-cubes</it>). Reaping fresh analytic results, we find that K-cubes minimize the wirecount and, <it>simultaneously</it>, the latency of hypercube architectures that tolerate failure of any <tmath>f</tmath> nodes. Refining the graph model of Hayes (1976), we pose the feasibility of configuration as a problem in multivariate optimization:</p><p>What <tmath>(f + 1){\hbox{-}}{\rm connected}</tmath><tmath>n{\hbox{-}}{\rm vertex}</tmath> graphs with fewest edges <tmath>\lceil n ( f + 1) / 2\rceil</tmath> minimize the maximum a) radius or b) diameter of subgraphs (i.e., <it>quorums</it>) induced by deleting up to <tmath>f</tmath> vertices? (1)</p><p>We solve (1) for <tmath>f</tmath> that is superlogarithmic but sublinear in <tmath>n</tmath> 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 <it>optimal</it>, while C-cubes are <it>suboptimal</it>. 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 <it>only</it> 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.</p>
Hypercube fault tolerance, hypercube latency, configuration architectures, performability, quorums, Hamming graphs, K-cubes, Moore graphs, Moore Bound, C-cubes, Lee distance.

M. S. Fadali, K. F. Korver and L. E. LaForge, "What Designers of Bus and Network Architectures Should Know about Hypercubes," in IEEE Transactions on Computers, vol. 52, no. , pp. 525-544, 2003.
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