The -Diagnosability for Regular Networks

Abstract

The $t/k$ -diagnosis strategy can significantly enhance the system’s self-diagnosing capability at the expense of no more than $k$ fault-free processors (vertices) being mistakenly diagnosed as faulty under the PMC model. It is a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis on multiprocessor systems. It can detect up to $t$ faulty processors (vertices) which might include at most $k$ misdiagnosed processors (vertices), where $k$ is typically a small number. In the case $k\ge 1$ , to our knowledge, there is no known $t/k$ -diagnosis algorithm for general regular networks. In this paper, we first propose a general $t/k$ -diagnosis ( $k\ge 1$ ) algorithm for some $m$ -regular networks. These $m$ -regular networks satisfying some conditions could establish the $t/k$ -diagnosis algorithm, say $t/k$ - $G$ - $DIAG$ , to determine the $t/k$ -diagnosability. The complexity of this algorithm is only $O(N\log N)$ (when $N\ge 2^m$ ) or $O(Nm)$ (when $N ) where $N$ is the number of vertices in the network. Second, we present a complete proof that the network $G$ is actually $t/k$ -diagnosable. Finally, we establish the $t/k$ -diagnosability ( $1\le k\le 3$ ) of some regular networks, including an $n$ -dimensional alternating group graph , an $n$ -dimensional Split-Star Network , a $l^n$ -hypermesh and an $(n,l)$ -star graph , which are well-known interconnection networks proposed for multiprocessor systems.