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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)-connected n-vertex graphs with fewest edges n(f+1)/2 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: I) the fault tolerance of K-cubes is proportionally greater than that of C-cubes; II) quorums formed from K-cubes have a diameter that is asymptotically convergent to the Moore Bound on radius; III) 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: IV) counterexamples, corrects, and generalizes a mistaken claim by Armstrong and Gray (1981) concerning binary cubes; V) 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 VI) 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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Introducing new analytic results, we minimize the cost of point-to-point fault tolerant avionics architectures. Refining the graph model of Hayes [23], we formulate the worst-case feasibility of configuration as:What Cf+l)-connected n-vertex graphs with fewest edges minimize the maximum radius or diameter4 of subgraphs (i.e., quorums) induced by deleting up to f of the n vertices?We solve this problem by proving: i) K-cubes (cubes based on cliques) can tolerate a greater proportion of faults than can traditional C-cubes (cubes based on cycles); ii) quorums formed from K-cubes have a diameter that is asymptotically equal to the Moore bound, while under no conditions of scaling can the Moore bound be attained by C-cubes whose radix exceeds 4. Thus, for fault tolerance logarithmic in n, K-cubes are optimal, whereas C-cubes are suboptimal. Our exposition also corrects and generalizes a mistaken claim by Armstrong and Gray [ 191 concerning binary cubes.

Local sparing is a simple way to organize the redundancy of a fault tolerant system. A n y system can be locally spared. Furthermore, local sparing preserves both regularity and planarity. In spite of this, the potential usefulness of local sparing appears to have been overlooked. Suppose that the designer wishes to assure, with high probability, a fault-free copy of the n-element system desired. If local sparing is used then, as we prove, i) the resulting area is e( log n ) times the area of the system desired; ii) the wirelength is O( +) times the maximum wirelength in the desired system; iii) an optimal diagnosis algorithm identifies the faulty elements in @ ( n log' n ) time; iv) in optimal time @(n log n + number or wires in the desired system) , a simple configuration algorithm achieves a fault-free copy of the desired system if and only if a fault-free copy exists. We illustrate these results for arrays, binary trees, and hypercubes. In addition, v) if Y denotes the probability of achieving a fault-free copy of the system desired then, usin6 h-fold redundancy, the maximum rate at which elements can fail is ( Local sparing is simple, widely-applicable, and low-cost. A disadvantage is that, depending on the system desired, the cost may not be optimal. However, there is strong reason to prefer local sparing over global sparing, and in some cases local sparing is better than more popular approaches to configuration.