A linear time pessimistic one-step diagnosis algorithm for hypercube multicomputer systems

Dong, Ting

doi:10.1016/j.parco.2005.05.002

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…where the last equality follows from Equation (5). Since σ (T d ) is the sum of total distances in the 0-tree, 1-tree, and the total distance between all vertices in the 0-tree and all vertices in the 1-tree, σ (T d ) can be expressed by the recurrence…”

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confidence: 99%

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On the average distance of the hypercube tree

Alsuwaiyel

2010

International Journal of Computer Mathematics

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Given a graph G on n vertices, the total distance of G is defined as σ G = (1/2) u,v∈V (G) d (u, v), where d(u, v) is the number of edges in a shortest path between u and v. We define the d-dimensional hypercube tree T d and show that it has a minimum total distance σ (T d ) = 2σ (H d ) − n 2 = (dn 2 /2) − n 2 over all spanning trees of H d , where H d is the d-dimensional binary hypercube. It follows that the average distance of T d is μ(T d ) = 2μ(H d ) − 1 = d(1 + 1/(n − 1)) − 1.

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Section: Downloaded By [Mcmaster University] At 00:02 20 December 2014mentioning

confidence: 99%

“…The names 'completely unbalanced spanning tree'(CUST) [21] and 'hypercube tree'also appeared in the contexts of fault-tolerant computing and diagnosis of hypercube multicomputer systems to isolate faulty processors [5].…”

Section: Introductionmentioning

confidence: 99%

On the average distance of the hypercube tree

Alsuwaiyel

2010

International Journal of Computer Mathematics

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“…The precise diagnosis and pessimistic diagnosis, i.e., k =0,1, have received a great deal of researches. () Yang and Tang presented a (4 n −9)/3 diagnosis algorithm for hypercubes. In the case of k ≥4, an efficient t / k diagnosis algorithm of hypercubes or hypercube‐like networks remains yet to be developed.…”

Section: Introductionmentioning

confidence: 99%

A t/k diagnosis algorithm on hypercube‐like networks

Xie

Liang

2017

Concurrency and Computation

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SummaryProcessor fault diagnosis takes a key role in fault-tolerant computing on multiprocessor systems.The t∕k diagnosis strategy which is a generalization of the precise and pessimistic diagnosis strategies can significantly improve the self-diagnosing capability of the system. Using this tool, it is possible to deal with large faults in the system. This paper presents a t∕k diagnosis algorithm on n-dimensional hypercube-like networks (include Hypercubes, Crossed cubes, Möbius cubes, Locally Twisted cubes, and Twisted cubes) for any k ∈ [0, n − 2]. The algorithm can correctly identify all nodes except at most k nodes undiagnosed. It runs in O(N) time, where N = 2 n is the total number of nodes of n-dimensional hypercube-like networks. To the best of our knowledge, in the case k ≥ 4, there is no known t∕k diagnosis algorithm for general diagnosable system or any specific system. KEYWORDShypercube-like networks, large connected component, system-level diagnosis, t∕k diagnosis strategy INTRODUCTIONWith the rapid development of very large scale integration (VLSI) technology, a multiprocessor system may contain hundreds or even thousands of processors (nodes). As the number of processors increases, so does the expected number of processors being faulty at the same time. To ensure reliability, the system should have the ability to identify the faulty processors which are then isolated from the system or replaced by additional fault-free ones. The technique of identifying faulty processors by conducting tests on the processors and interpreting the test outcomes is known as system-level diagnosis.Preparata et al. 1 proposed the first model of system-level diagnosis, known as the PMC model. It assumes that each node in a system can conduct tests on its neighbors via the communication links between them. When a node tests its neighbor, the tester declares the tested node to be fault-free or faulty based on the test responses. This evaluation is reliable if and only if the testing node is fault-free.There are two classical diagnosis strategies, i.e., the precise diagnosis strategy 1 and the pessimistic diagnosis strategy. 2,3 However, due to their limited diagnosability, they are unable to deal with large faults in large multiprocessor systems. In order to greatly enhance the degree of diagnosability, Somani and Peleg proposed a new diagnosis strategy known as the t∕k diagnosis. 4 Under this strategy, all the faulty nodes can be isolated to within a set in which at most k nodes is fault-free. Specifically, a system S of N nodes is t∕k-diagnosable if, given any test syndrome produced by the system under the presence of a fault set F, all the faulty nodes can be isolated to within a set F ′ , out of which at most k nodes can possibly be fault-free, |F ′ | ≤ |F| + k, provided the number of faulty nodes does not exceed t. By definition, if k = 0, the t∕k diagnosis reduces to the precise diagnosis; if k = 1, the t∕k diagnosis is closely related to the pessimistic diagnosis. By choosing a larger k, the t∕k diagnosis can signif...

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“…It has been extensively studied (see [5] for an excellent survey and results). Applications of the binary hypercube tree in broadcasting and personalized communication and in fault-tolerant computing can be found in [1,6,10,11,14]. In [16], it is shown that the binary hypercube tree is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H 2 (d), attempts to improve the average distance between pairs of nodes, by adding an edge e of H 2 (d) − T and removing an edge e from the (unique) cycle created by e. In [2], it is shown that the hypercube tree has the minimum total distance among all spanning trees.…”

Section: Introductionmentioning

confidence: 99%

On thek-ary hypercube tree and its average distance

Alsuwaiyel

2011

International Journal of Computer Mathematics

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To cite this article: M. H. Alsuwaiyel (2011) On the k-ary hypercube tree and its average distanceThe d-dimensional k-ary hypercube tree T k (d) is a generalization of the hypercube tree, also known in the literature as the spanning binomial tree. We present some of its structural properties and investigate in detail its average distance. For instance, it is shown that the binary hypercube tree has the anomaly of having two nodes in its centre as opposed to having one in hypercube trees of arity k > 2. However, in all dimensions, the centre, centroid and median coincide. We show that its total distance is σ (T k whose limiting value is 2. This answers a generalization of a conjecture of Dobrynin et al. [Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), pp. 211-249] for the binary hypercube by the affirmative.

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A linear time pessimistic one-step diagnosis algorithm for hypercube multicomputer systems

Cited by 6 publications

References 14 publications

On the average distance of the hypercube tree

On the average distance of the hypercube tree

A t/k diagnosis algorithm on hypercube‐like networks

On thek-ary hypercube tree and its average distance

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