Fault diagnosability of arrangement graphs

The growing size of the multiprocessor system increases its vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain a system's high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. This paper shows that the largest connected component of the survival graph contains almost all of the remaining vertices in the dual-cube DCnwhen the number of faulty vertices is up to twice or three times of the traditional connectivity. Based on this fault resiliency, this paper determines that the conditional diagnosability of DCn(n ≥ 3) under the comparison model is 3n − 2, which is about three times of the traditional diagnosability.

The conditional fault diagnosability of (n, k)-star graphs

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 $t/m$ -diagnosability are two important diagnostic strategies at system-level 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 $t/m$ -diagnosis 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 $(2k-1) (n-k)-1$ ( $k\geq 3$ , $n\geq k+2$ ), two-extra connectivity is $(3k-2)(n-k)-3$ ( $k\geq 4$ , $n\geq k+2$ ), and three-extra connectivity is $(4k-4)(n-k)-4$ ( $k\geq 4$ , $n\geq k+2$ or $k\geq 3$ , $n\geq k+3$ ), respectively. And then, we address the $g$ -extra conditional diagnosability of $A_{n,k}$ under the PMC model for $1\leq g \leq 3$ . Finally, we determine that the $(n,k)$ -arrangement graph $A_{n,k}$ is $[(2k-1)(n-k)-1]/1$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), $[(3k-2)(n-k)-3]/2$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), and $[(4k-4)(n-k)-4]/3$ -diagnosable ( $k\geq 4$ , $n\geq k+3$ ) under the PMC model, respectively.

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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(n-1,k-1)$</tex></formula> -subgraph in the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(n,k)$</tex></formula> -Arrangement Graph <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$A_{n,k}$</tex> </formula> , 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> goes low.

The growing size of a multiprocessor system increases its vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain a system’s high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. Through fault tolerance analysis of the multiprocessor system based on twisted-cube connected network TN n , we derive the conditional diagnosability of the system, which is about three times of its classical diagnosability under the comparison model. KeywordsConditional diagnosabilityComparison diagnosis modelTwisted-cube connected network

The growing size of the multiprocessor systems increases their vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain the system's high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. The conditional diagnosis requires that for each processor v in a system, all the processors that are directly connected to v do not fail simultaneously. In this paper, we show that the conditional diagnosability of the crossed cubes CQ n under the comparison diagnosis model is 3n−5 when n≥7. Hence, the conditional diagnosability of CQ n is three times larger than its classical diagnosability.

The growing size of the multiprocessor systems increases their vulnerability to component failures. It is crucial to locate and to replace the faulty processors to maintain system's high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. The conditional diagnosis requires that for each processor v in a system, all the processors that are directly connected to v do not fail at the same time. In this paper, the conditional diagnosability of the locally twisted cubes LTQ n under the comparison diagnosis model is 3n-5 when n≫6. Hence the conditional diagnosability of LTQ n is three times larger than its classical diagnosability.

The design of large dependable multiprocessor systems requires quick and precise mechanisms for detecting the faulty nodes. The system-level fault diagnosis is the process of identifying faulty processors in a system through testing. This paper shows that the largest connected component of the survival graph contains almost all remaining vertices in the hierarchical hypercube HHC n when the number of faulty vertices is up to two or three times of the traditional connectivity. Based on this fault resiliency, we establish that the conditional diagnosability of HHC n (n=2 m +m, m≥2) under the comparison model is 3m−2, which is about three times of the traditional diagnosability.

The star graph proposed by Akers et al. (Proc Int Conf Parallel Process, University Park, PA, 1987, pp. 393–400) has many advantages over the n‐cube. However, it suffers from having large gaps in the possible number of vertices. The arrangement graph was proposed by Day and Tripathi (Inf Process Lett 42 (1992), 235–241) to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. In fact, it also generalizes the alternating group graph (Jwo et al., Networks 23 (1993), 315–326). There are many different measures of structural integrity of interconnection networks. In this article, we prove results of the following type for the arrangement graph: If h(r,n,k) vertices are deleted from the arrangement graph An,k, the resulting graph will either be connected or have a large component and small components having at most r − 1 vertices in total. Our result is tight for r ≤ 3, and it is asymptotically tight for r ≥ 4. Moreover, we also determine the cyclic vertex‐connectivity of the arrangement graph. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Diagnosability of the Incomplete Star Graphs

The pessimistic diagnosability of three kinds of graphs

The growing size of multiprocessor systems increases the vulnerability to component failures. It is crucial to locate and replace faulty processors to maintain the system's high reliability. Processor fault diagnosis is essential to the reliability of a multiprocessor system and the diagnosabilities of many well-known networks (such as hierarchical hypercubes and crossed cubes [S. Zhou, L. Lin and J.-M. Xu, Conditional fault diagnosis of hierarchical hypercubes, Int. J. Comput. Math. 89(16) (2012), pp. 2152–2164 and S. Zhou, The conditional diagnosability of crossed cubes under the comparison model, Int. J. Comput. Math. 87(15) (2010), pp. 3387–3396]) have been investigated in the literature. A system is t-diagnosable if all faulty nodes can be identified without replacement when the number of faults does not exceed t, where t is some positive integer. Furthermore, a system is strongly t-diagnosable if it is t-diagnosable and can achieve (t+1)-diagnosability except for the case where a node's neighbours are all faulty. In addition, conditional diagnosability has been widely accepted as a new measure of diagnosability by assuming that any fault-set cannot contain all neighbours of any node in a multiprocessor system. In this paper, we determine the conditional diagnosability and strong diagnosability of an n-dimensional shuffle-cube SQn, a variant of hypercube for multiprocessor systems, under the comparison model. We show that the conditional diagnosability of shuffle-cube SQn (n=4k+2 and k≥2) is 3n−9, and SQn is strongly n-diagnosable under the comparison model.

We derive an explicit formula for the surface area of the arrangement graph, i.e. the number of vertices at a certain distance from the identity vertex in such a graph. We also present such formulas for the star graph, the alternating group graph, and the split-star graph, via their respective structural relationship to the arrangement graph.

All approach that uses a hardware device to perform fault diagnosis in a fully distributed multiprocessor system is proposed. Both the complexities of the diagnostic hardcore and interprocessor communication are reduced. Fault diagnosis in a fully distributed system with permanent faults is considered. Standard notation, definitions, and a framework for discussion are provided. A brief description of the general fault diagnosis process follows. The structure of the diagnostic hardware for each component in the system, called a component diagnosis element (CDE), is defined. The input dependency of each CDE is presented along with a discussion of how this diagnosis device carries out fault diagnosis in a distributed environment. The necessary conditions are provided for a class of fully distributed systems where faulty processors can be diagnosed independently within each processor provided that the number of faulty processors does not exceed a set limit.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A network's diagnosability is the maximum number of faulty vertices the network can discriminate solely by performing mutual tests among the vertices. It is an important measure of a network's robustness. The original diagnosability without any condition is often rather low because it is bounded by the network's minimum degree. Several conditional diagnosability have been proposed in the past to increase the allowed faulty vertices, and hence enhancing the diagnosability of the network. The g-good-neighbor conditional diagnosability is the maximum number of faulty vertices a network can guarantee to identify, under the condition that every fault-free vertex has at least g fault-free neighbors (i.e., good neighbors). In this paper, we establish the g-good-neighbor conditional diagnosability for the (n; k)-arrangement graph network A n;k . We will show that, under both the PMC model and the comparison model, the A n;k 's g-good-neighbor conditional diagnosability is [(g + 1)k - g](n - k), which can be several times higher than the A n;k 's original diagnosability.