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Issue No.04 - April (2003 vol.52)

pp: 525-544

Laurence E. LaForge , IEEE

Kirk F. Korver , IEEE

M. Sami Fadali , IEEE

ABSTRACT

<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>

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