Many-to-many edge-disjoint paths in (n,k)-enhanced hypercube under three link-faulty hypotheses

Abstract

One of the most central issues in interconnection networks of parallel and distributed systems is finding edge-disjoint paths that transmit information. Finding as many as possible many-to-many edge-disjoint paths is conducive to improving the fault-tolerance of such networks. As an interconnection network topology, (n,k)-enhanced hypercube Qn,k (1≤k≤n−1), is a momentous variant of well-known hypercube. For integers 1≤l≤n−1 and n≥2, let δ=0 if 1≤l≤n−k, and δ=1 if n−k+1≤l≤n−1. This paper offers a unified method to determine the minimum cardinalities of faulty links in Qn,k, whose malfunction divides this network into several connected components such that each processor has at least l+δ neighbors, each component contains no less than 2l processors and the number of average neighbors for all processors is at least l+δ, respectively. Under these three different link-faulty assumptions, but for the condition of k=2 and l=n−2, the minimum cardinalities of such faulty links share the same value (n−l−δ+1)2l. And the value in the exceptional case is (n−l−δ)2l+1=2n−1. In other words, we find the maximum numbers of many-to-many edge-disjoint paths of Qn,k under the above three hypotheses, which offers refined measurements for the reliability and fault-tolerance of interconnection networks.