Fault diagnosability of arrangement graphs

Hsieh

2016

IEEE Trans. Rel.

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Extra connectivity is an important indicator of the robustness of a multiprocessor system in presence of failing processors. The g-extra conditional diagnosability and the t/mdiagnosability are two important diagnostic strategies at systemlevel that can significantly enhance the system's self-diagnosing capability. The g-extra conditional diagnosability is defined under the assumption that every component of the system removing a set of faulty vertices has more than g vertices. The t/mdiagnosis strategy can detect up to t faulty processors which might include at most m misdiagnosed processors, where m is typically a small integer number. In this paper, we analyze the combinatorial properties and fault tolerant ability for an (n, k)-arrangement graph, denoted by A n,k , a well-known interconnection network proposed for multiprocessor systems. We first establish that the A n,k 's one-extra connectivity is (2k − 1) 3 (k ≥ 4, n ≥ k + 2), and three-extra connectivity is (4k − 4)(n − k) − 4 ( k ≥ 4, n ≥ k + 2 or k ≥ 3, n ≥ k + 3), respectively. And then, we address the g-extra conditional diagnosability of A n,k under the PMC model for 1 ≤ g ≤ 3. Finally, we determine that the (n, k)-arrangement graph A n,k is [(2k − 1)(n − k) − 1]/1-diagnosable (k ≥ 4, n ≥ k + 2), [(3k − 2)(n − k) − 3]/2-diagnosable (k ≥ 4, n ≥ k + 2), and [(4k − 4)(n − k) − 4]/3-diagnosable (k ≥ 4, n ≥ k + 3) under the PMC model, respectively.

“…. Fault tolerant routing and conditional diagnosability of arrangement graphs have been widely investigated [5], [6], [8], [22]- [24], [38], [39].…”

Section: Arrangement Graphs a Nkmentioning

confidence: 99%

“…For simplicity in context, by the symmetry of A n,k , we shall take j as the last position k, and use A i n,k to denote [15], [16], [39]…”

Section: Arrangement Graphs a Nkmentioning

confidence: 99%

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The Extra Connectivity, Extra Conditional Diagnosability, and <inline-formula> <tex-math notation="LaTeX"> $t/m$</tex-math> </inline-formula>-Diagnosability of Arrangement Graphs

Hsieh

2016

IEEE Trans. Rel.

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“…Chiang and Chen [4] showed that is isomorphic to the -alternating group graph . Recent years have witnessed the continued interest in the research of arrangement graphs [8], [9], [13], [17], [19], [24], [25].…”

Section: Index Terms-arrangement Graph Inclusion-exclusion Prinmentioning

confidence: 99%

The Reliability of Subgraphs in the Arrangement Graph

Lin

et al. 2015

IEEE Trans. Rel.

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As the size of a multiprocessor computer system grows, the probability of having faulty (i.e., malfunctioning or failing) processors in the system increases. It is then important to quantify how the faults collectively affect the entire system. The reliability of subsystems in a system, defined as the probability that a fault-free subsystem of a certain size still exists when the system has faults, is a measure for the faults' effect on the whole system. It can be used as an indicator of system health. In this paper, we will present two schemes to calculate the reliability of an -subgraph in the -Arrangement Graph , an extensively studied interconnection network proposed for multiprocessor computers. The first scheme will use a probability fault model and the Principle of Inclusion-Exclusion to establish an upper-bound of the reliability, by taking into account the intersection of not more than three subgraphs. The second scheme uses basically the same idea, but completely neglects the intersection among subgraphs to calculate an approximate reliability. The results of the two schemes are compared, and are shown to be in good agreement, especially as the single-node reliability goes low.

“…Through the evaluation of the size of connected components [9], [10], Lin et al [24] established that the conditional diagnosability of the Star Graph under the comparison model is , which is about three times larger than the diagnosability under the classical definition. Recently, Zhou and Xu [37] have obtained the conditional diagnosability for the Arrangement Graphs under the comparison model, of which the Star Graph is a special case. The diagnosabilities of Star Graphs with missing edges under the PMC model, BGM model and the comparison model have been investigated by Zheng and Zhou [36], and Chiang et al [11], respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Lin

et al. 2015

IEEE Trans. Comput.

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The -diagnosis is a diagnostic strategy at system level that can significantly enhance the system's self-diagnosing capability. It can detect up to faulty processors (or nodes, units) which might include at most misdiagnosed processors, where is typically a small number. Somani and Peleg ([26], 1996) claimed that an -dimensional Star Graph (denoted ), a well-studied interconnection model for multiprocessor systems, is -diagnosable. Recently, Chen and Liu ([5], 2012) found counterexamples for the diagnosability obtained in [26], without further pursuing the cause of the flawed result. In this paper, we provide a new, complete proof that an -dimensional Star Graph is actually -diagnosable, where , and investigate the reason that caused the flawed result in [26]. Based on our newly obtained fault-tolerance properties, we will also outline an diagnostic algorithm ( is the number of nodes in ) to locate all (up to ) faulty processors, among which at most fault-free processors might be wrongly diagnosed as faulty.