A unified approach to reliability and edge fault tolerance of cube-based interconnection networks under three hypotheses

Abstract

The topological structures of the interconnection networks of some parallel and distributed systems are designed as n-dimensional hypercube $Q_n$ or n-dimensional folded hypercube $FQ_n$ with $N=2^n$ processors. For integers $0\le k\le {n-1}$, let $\mathcal {P}_1^k$, $\mathcal {P}_2^k$ and $\mathcal {P}_3^k$ be the property of having at least k neighbors for each processor, containing at least $2^k$ processors and admitting average neighbors at least k, respectively. $\mathcal {P}$-conditional edge-connectivity of G, $\lambda (\mathcal {P},G)$, is the minimum cardinality of faulty edge-cut, whose malfunction divides this network into several components, with each component satisfying the property of $\mathcal {P}$. For each integer $0\le k\le {n-1}$, and $1\le i\le 3$, this paper offers a unified method to investigate the $\mathcal {P}_i^k$-conditional edge-connectivity of $Q_n$ and $FQ_n$. Exact value of $\mathcal {P}_i^k$-conditional edge-connectivity of $Q_n$, $\lambda (\mathcal {P}_i^k,Q_n)$, is $(n-k)2^k$, and that of $\mathcal {P}_i^k$-conditional edge-connectivity of $FQ_n$, $\lambda (\mathcal {P}_i^k,FQ_n)$, is $(n-k+1)2^k$. Our method generalizes the result of Guo and Guo in [The Journal of Supercomputing, 2014, 68:1235-1240] and the previous other results.