Design of gracefully degradable hypercube-connected systems

Abstract

We address the problem of modifying a hypercube computer by the addition of spare nodes and links to improve its fault tolerance, while maintaining a specified level of performance. The hypercube is modeled by a graph in which nodes represent processors and edges represent communication links. A new graph-based measure of performance degradation is introduced. This characterizes a fault-tolerant hypercube as k-fault-tolerant (k-FT) g-step-degradable (g-SD) if the removal of any k nodes reduces the dimension of the largest fault-free subcube by at most g. We show how to construct k-FT g-SD hypercubes for values of k up to 16 and g = 0, 1, or 2. Many of these designs are shown to be link- or degree-optimal. We also propose a construction method that uses small k-FT g-SD designs as seeds to construct k-FT g-SD designs of larger sizes. This results in fault-tolerant hypercubes in which reconfiguration can be first done locally and then easily extended to the entire system. The small number of added links and nodes is shown to be useful not only in increasing the fault tolerance of the underlying hypercube, but also in reducing the average internode distance.