Shuming Zhou

Conditional diagnosability of alternating group networks

The growing size of a multiprocessor system increases its vulnerability to component failures. In order to maintain the systems high reliability, it is crucial to identify and replace the faulty processors through testing, a process known as fault diagnosis. The minimum size of a largest connected component in such a networked system is typically used as a measure for fault tolerance of the system. For this measure, the conditional diagnosability of the system in terms of an alternating group network is important, which is studied in the present paper under a comparison model, with some precise and useful bounds of tolerance derived.

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 conditional diagnosability of crossed cubes under the comparison model

In evaluating the fault tolerance of an network structure, it is essential to estimate the order of a maximal connected component of this network provided the faulty vertices may break its connectedness, and it is crucial to local and to replace the faulty processors to maintain system’s high reliability. The fault diagnosis is the process of identifying fault 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 twisted cubes TQn under the comparison diagnosis model is 3n-5 when n≫6. Hence the conditional diagnosability of TQn is three times larger than its classical diagnosability.

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.

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

In cloud storage, storage servers may not be fully trustworthy. Therefore, it is of great importance for users to check whether the data is kept intact. This is the goal of remote data possession checking (RDPC) schemes. In this paper, an RDPC scheme based on homomorphic hashing is proposed. To enable data dynamics, the Merkle hash tree is introduced to record the location for each data operation in the scheme. Data dynamics, including the most general forms of data operations such as block modification, insertion and deletion, are supported. Our scheme provides provable data possession and integrity protection. The security and performance analysis shows that the scheme is practical for real-world use.

-Diagnosability of Star Graph Networks

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.

Data dynamics for remote data possession checking in cloud storage

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 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 DCn when 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.

Conditional fault diagnosis of hierarchical hypercubes

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.