Conditional diagnosability and strong diagnosability of shuffle-cubes under the comparison model

Abstract

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 SQ n , a variant of hypercube for multiprocessor systems, under the comparison model. We show that the conditional diagnosability of shuffle-cube SQ n (n = 4k + 2 and k ≥ 2) is 3n − 9, and SQ n is strongly n-diagnosable under the comparison model.

The Extra Connectivity, Extra Conditional Diagnosability, and <inline-formula> <tex-math notation="LaTeX"> $t/m$</tex-math> </inline-formula>-Diagnosability of Arrangement Graphs

“…Since it is impossible that all neighbors of some processor are simultaneously faulty, Zhang and Yang [36] proposed the gextra conditional diagnosability (defined in Section II), which is a generalization of conditional diagnosability [3], [19], [20]. 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.…”

Minimum Neighborhood of Alternating Group Graphs

“…We will further applied the minimum neighborhood of AG n to obtain the fault tolerance of the n-split-star network [18] in the future. Meanwhile, the minimum neighborhood can be applied to obtain all kinds of conditional fault tolerance and a variety of conditional diagnosability.…”

“…In 2009, Yang and Meng [31] established the size of a subgraph A of Q n which is (q + 1)n − 2q − q(q − 1)/2 when |V (A)| = q + 1 and used it to obtain the extra connectivity. In 2015, Lin et al [18] proposed the size of neighbors of a subset D in n-shuffle-cube, which can be applied to conditional diagnosability. In 2016, Zhao et al [39] proposed that the size of neighbors of an independent set D of vertices with |D| = q in Q n is ≥ −q 2 /2 + (2n − 5/2)q − n 2 +2n+1 for n+1 ≤ q ≤ 2n−4 and applied it to obtain the component connectivity.…”

Symmetric properties and two variants of shuffle-cubes

“…The reliability of the shuffle cube concerning refined connectivity were determined in [11,29,33]. The conditional diagnosability of SQ n was studied in [23,34]. The matching preclusion number was determined by Antantapantula et al [4].…”