The $t/k$-Diagnosability of Star Graph Networks

Abstract

The ${{t/k}}$ -diagnosis is a diagnostic strategy at system level that can significantly enhance the system’s self-diagnosing capability. It can detect up to ${{t}}$ faulty processors (or nodes, units) which might include at most ${{k}}$ misdiagnosed processors, where ${ {k}}$ is typically a small number. Somani and Peleg ( , 1996) claimed that an $n$ -dimensional Star Graph (denoted ${{S_n}}$ ), a well-studied interconnection model for multiprocessor systems, is ${{((k + 1)n - 3k - 2)/k}}$ -diagnosable. Recently, Chen and Liu ( , 2012) found counterexamples for the diagnosability obtained in , without further pursuing the cause of the flawed result. In this paper, we provide a new, complete proof that an ${\mbi {n}}$ -dimensional Star Graph is actually ${{((k + 1)n - 3k - 1)/k}}$ -diagnosable, where ${{1 \leq k \leq 3}}$ , and investigate the reason that caused the flawed result in . Based on our newly obtained fault-tolerance properties, we will also outline an ${ {O(N \log N)}}$ diagnostic algorithm ( ${ {N = n!}}$ is the number of nodes in ${{S_n}}$ ) to locate all (up to ${ {(k + 1)n - 3k - 1}}$ ) faulty processors, among which at most ${ {k\, (1 \leq k \leq 3)}}$ fault-free processors might be wrongly diagnosed as faulty.