Limei Lin

The Extra Connectivity and Conditional Diagnosability of Alternating Group Networks

Extra connectivity, diagnosability, and conditional diagnosability are all important measures for a multiprocessor systems ability to diagnose and tolerate faults. In this paper, we analyze the fault tolerance ability for the alternating group graph, a well-known interconnection network proposed for multiprocessor systems, establish the h-extra connectivity, where 1 ≤ h ≤ 3, and prove that the conditional diagnosability of an n-dimensional alternating group graph, denoted by AGn, is 8n - 27 (n ≥ 4) under the PMC model. This is about four times of the AGns traditional diagnosability. As a byproduct, the strong diagnosability of AGn is also obtained.

The Extra, Restricted Connectivity and Conditional Diagnosability of Split-Star Networks

Connectivity is a classic measure for fault tolerance of a network in the case of vertices failures. Extra connectivity and restricted connectivity are two important indicators of the robustness of a multi-processor system in presence of failing processors. An interconnection networks diagnosability is an important measure of its self-diagnostic capability. The conditional diagnosability is widely accepted as a new measure of diagnosability by assuming that any fault-set cannot contain all neighbors of any node in a multiprocessor system. In this paper, we analyze the combinatorial properties and fault tolerance ability for the Split-Star Network, denoted by Sn2, a well-known interconnection network proposed for multiprocessor systems, establish the g-extra connectivity, where 1 ≤ g ≤ 3. We also determine the h-restricted connectivity (h = 1; 2), and prove that the conditional diagnosability of Sn2 (n ≥ 4) is 6n - 16 under the comparison model, which is about three times of the Sn2s traditional diagnosability. As a product, the strong diagnosability of Sn2 is also obtained.

Conditional diagnosability of arrangement graphs under the PMC model

Abstract Processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system, and the diagnosabilities of many well-known multiprocessor systems have been investigated. The conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault-set cant contain all the neighbors of any node in a multiprocessor system. This paper considers the conditional diagnosability of an ( n , k ) -arrangement graph A n , k , a flexible interconnection network model for multiprocessor systems, under the classical PMC diagnostic model, and determines that the conditional diagnosability of A n , k ( k ≥ 2 , n ≥ k + 2 ) is ( 4 k − 4 ) ( n − k ) − 3 , which is about four times its traditional diagnosability.

The

The t/k-diagnosis is a diagnostic strategy at system level that can significantly enhance the systems 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 ([26], 1996) claimed that an n-dimensional Star Graph (denoted Sn), a well-studied interconnection model for multiprocessor systems, is ((k + 1)n - 3k - 2)/k-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 n-dimensional Star Graph is actually ((k + 1)n - 3k - 1)/k-diagnosable, where 1 ≤ k ≤ 3, and investigate the reason that caused the flawed result in [26]. 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 Sn) to locate all (up to (k + 1)n - 3k - 1) faulty processors, among which at most k (1 ≤ k ≤ 3) fault-free processors might be wrongly diagnosed as faulty.

t/k

An interconnection networks diagnosability is an important measure of its self-diagnostic capability. The classical problems of fault diagnosis are explored widely. The conditional diagnosability is proposed by Lai et al. as a new measure of diagnosability, which can better measure the diagnosability of regular interconnection networks. The conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors. Furthermore, a multiprocessor system is strongly t-diagnos-able, if it is t-diagnosable and can achieve diagnosability t + 1 except for the case where a nodes neighbors are all faulty. The conditional diagnosability and strong diagnosability were proposed later to better reflect the networks self-diagnostic capability under more realistic assumptions. In this paper, we determine the conditional diagnosability of an n-dimensional Split-Star Network (denoted as S n 2 ), a well-known interconnection network model for multiprocessor systems, under the PMC (Preparata, Metze, and Chien) model. We show that the conditional diagnosability of S n 2 ( n ? 4 ) is 8 n - 23 , which is about four times of its traditional diagnosability. As a byproduct, the strong diagnosability of S n 2 is also obtained. We investigate the combinatorial properties and fault-tolerant properties of the Split-Star Network.We show that as long as a fault-set F has 6 n - 17 or fewer nodes, S n 2 ? F , excluding a subset of at most two nodes, contains one large connected component.We show that as long as a fault-set F has 8 n - 25 or fewer nodes, S n 2 ? F , excluding a subset of at most three nodes, contains one large connected component.We show that the classic diagnosability of S n 2 ( n ? 2 ) under the PMC model is 2 n - 3 .We also establish that the conditional diagnosability and the strong diagnosability of S n 2 are 8 n - 23 and 2 n - 3 under the PMC model.

-Diagnosability of Star Graph Networks

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 (n-1,k-1)-subgraph in the (n,k)-Arrangement Graph An,k, 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 p goes low.

Conditional diagnosability and strong diagnosability of Split-Star Networks under the PMC model

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 systems 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 An,k, a well-known interconnection network proposed for multiprocessor systems. We first establish that the An,ks one-extra connectivity is (2k - 1) (n - k) - 1 (k ≥ 3, n ≥ k + 2), two-extra connectivity is (3k - 2)(n - k) - 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 An,k under the PMC model for 1 ≤ g ≤ 3. Finally, we determine that the (n, k)-arrangement graph An,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.

The Reliability of Subgraphs in the Arrangement Graph

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 Extra Connectivity, Extra Conditional Diagnosability, and t/m-Diagnosability of Arrangement 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 systems 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 nodes 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.

Conditional fault diagnosis of hierarchical hypercubes

As the cardinality of multiprocessor systems grows, the probability of arising malfunctioning or failing processors in the system is bound to increase. It is then of both practical and theoretical importance to know the reliability of the system as a whole. One metric for a systems overall reliability is the measurement of the collective effect of its subsystems becoming faulty. However, a challenge of this approach is that the subsystems often interact with each other in a complex manner, making the analysis difficult. Wu and Latifi (Int. Sci., vol. 178, pp. 2337-2348, Oct. 2008) proposed two schemes to evaluate the system reliability of the Star graph network under a probabilistic fault model. The first scheme computes the combinatorial probability of subgraphs to obtain an upper-bound on the reliability by considering the intersection of no more than three subgraphs. The second scheme computes an approximate combinatorial probability by completely neglecting the intersection among subgraphs. Recently, Lin et al. have applied this approach to investigate the reliability of the multiprocessor system based on the arrangement graph (IEEE Trans. Rel., vol. 62, no. 2, pp. 807-818, Jun. 2015). In this paper, we extend the above approach by computing both upper- and lower-bounds and considering the difference of the two, to establish the reliability of the (n, k) -Star graph, another extensively studied interconnection network for multiprocessor systems. More specifically, we compute a lower-bound and an upper-bound on the reliability by taking into account the intersection of no more than four or three subgraphs, respectively. The empirical study shows that the upper- and lower-bounds are both very close to the approximate results. Especially, the lower the single-node reliability goes, the closer the approximate reliability is to both lower- and upper-bounds.