Subcube Fault-Tolerance in Hypercubes

Abstract

We consider the problem of determining the minimum number of faulty processors, K(n, m), and of faulty links, [lambda](n, m), in an n-dimensional hypercube computer so that every m-dimensional subcube is faulty. Best known lower bounds for K(n, m) and [lambda](n, m) are proved, several new recursive inequalities and new upper bounds are established, their asymptotic behavior for fixed m and for fixed n - m is analyzed, and their exact values are determined for small n and m. Most of the methods employed show how to construct sets of faults attaining the bounds. An extensive survey of related work is also included, showing connections to resource allocation, k-independent sets, and exhaustive testing.