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, λ( 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 λ( 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.