Random and Conditional (t, k)-Diagnosis of Hypercubes

Abstract

Under the PMC model, we consider the (t, k)-diagnosability of a hypercube under the random fault model and conditional fault model separately. A system is called random (t, k)-diagnosable [or conditionally (t, k)-diagnosable] if at least k faulty vertices that can be identified in each iteration under the assumption that there is no any restriction on the fault distribution (or every vertex is adjacent to at least one fault-free vertex), provided that there are at most t faulty vertices, where \(t\ge k\). When the remaining number of faulty vertices is lower than k, all of them can be identified. In this paper, under the PMC and random fault models, we show that the sequential t-diagnosis algorithm for hypercubes proposed by [35] can be extended to the (t, k)-diagnosis algorithm for hypercubes, and we show that the n-dimensional hypercubes are randomly \(\left( \genfrac(){0.0pt}0{n}{\frac{n}{2}},n\right) \)-diagnosable if n is even, and randomly \(\left( 2\cdot \genfrac(){0.0pt}0{n-1}{\frac{n-1}{2}},n\right) \)-diagnosable if n is odd, where \(\genfrac(){0.0pt}0{p}{q}=\frac{p!}{q!(p-q)!}\). Moreover, we propose a conditional (t, k)-diagnosis algorithm for hypercubes by using properties of the conditional fault model and show that n-dimensional hypercubes are conditionally \(\left( \genfrac(){0.0pt}0{n}{\frac{n}{2}},2n-2\right) \)-diagnosable if n is even, and conditionally \(\left( 2\cdot \genfrac(){0.0pt}0{n-1}{\frac{n-1}{2}},2n-2\right) \)-diagnosable if n is odd. Furthermore, under the PMC model, our results improve the previous best low bounds on t under the random and conditional fault models.