Connectivity of the crossed cube

SUMMARYIn this paper, we investigate the fault-tolerant Hamiltonian problems of crossed cubes with a faulty path. More precisely, let P denote any path in an n-dimensional crossed cube CQ n for n ≥ 5, and let V(P) be the vertex set of P. We show that CQ n − V(P) is Hamiltonian if |V(P)| ≤ n and is Hamiltonian connected if |V(P)| ≤ n−1. Compared with the previous results showing that the crossed cube is (n − 2)-fault-tolerant Hamiltonian and (n − 3)-fault-tolerant Hamiltonian connected for arbitrary faults, the contribution of this paper indicates that the crossed cube can tolerate more faulty vertices if these vertices happen to form some specific types of structures.

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“…It was proved that CQ n is n-connected [21] and its diameter is n+1 2 [11]. A vertex u = u n u n−1 .…”

Section: Definitionmentioning

confidence: 99%

The Fault-Tolerant Hamiltonian Problems of Crossed Cubes with Path Faults

Chen

Kung

Zou

et al. 2015

IEICE Trans. Inf. & Syst.

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“…Also, CQ n is regular of degree n and has connectivity n [16]; thus, by [6], CQ n has diagnosability n, when n ≥ 4.…”

Section: Hypercubes and Their Variantsmentioning

confidence: 99%

A general algorithm for detecting faults under the comparison diagnosis model

Stewart

2010

2010 IEEE International Symposium on Parallel &Amp; Distributed Processing (IPDPS)

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe develop a widely applicable algorithm to solve the fault diagnosis problem in certain distributed-memory multiprocessor systems in which there are a limited number of faulty processors. In particular, we prove that if the underlying graph G = (V, E) forming the interconnection network has connectivity no less than its diagnosability δ and can be partitioned into enough connected components of large enough size then given a syndrome of test results under the comparison diagnosis model resulting from some set of faulty nodes of size at most δ, we can find the actual set of faulty nodes with time complexity O(∆N ), where ∆ is the maximal degree of any node of the graph and N is the number of nodes.

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“…Lemma 6 [17] Suppose that (s, t) is a d-link of CQ n , where d is odd or d = n − 2 and n ≥ 2. Then (s (n−1) , t (n−1) ) is also a d-link of CQ n .…”

Section: Preliminariesmentioning

confidence: 99%

Embedding fault-free cycles in crossed cubes with conditional link faults

Hung

Chen

2008

J Supercomput

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The crossed cube, which is a variation of the hypercube, possesses some properties that are superior to those of the hypercube. In this paper, we show that with the assumption of each node incident with at least two fault-free links, an ndimensional crossed cube with up to 2n − 5 link faults can embed, with dilation one, fault-free cycles of lengths ranging from 4 to 2 n . The assumption is meaningful, for its occurrence probability is very close to 1, and the result is optimal with respect to the number of link faults tolerated. Consequently, it is very probable that algorithms executable on rings of lengths ranging from 4 to 2 n can be applied to an n-dimensional crossed cube with up to 2n − 5 link faults.

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Connectivity of the crossed cube

Cited by 73 publications

References 7 publications

The Fault-Tolerant Hamiltonian Problems of Crossed Cubes with Path Faults

The Fault-Tolerant Hamiltonian Problems of Crossed Cubes with Path Faults

A general algorithm for detecting faults under the comparison diagnosis model

Embedding fault-free cycles in crossed cubes with conditional link faults

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