Alternating Group Networks Research Articles

The pessimistic diagnosability of three kinds of graphs

A system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by tp(G), is the maximal number of faulty processors so that the system G is t/t-diagnosable. The pessimistic diagnosability of alternating group graphs AGn (Tsai, 2015); BC networks (Fan, 2005; Tsai, 2013); the k-ary n-cube networks Qnk, (Wang et al., 2012); regular graphs including the alternating group networks ANn (Hao et al., 2016) etc. But most of these results are about networks G with cn(G)≤2 (where cn(G) is the maximum number of common neighbors for any two distinct vertices). In this paper, we study the pessimistic diagnosability of three kinds of graphs which are (n,k)-arrangement graphs An,k, (n,k)-star graphs Sn,k and balanced hypercubes BHn, where cn(An,k)=cn(Sn,k)=n−k−1 and cn(BHn)=2n. We proved that tp(An,k)=(2k−1)(n−k)−1 for n≥k+2 and k≥3, tp(Sn,k)=n+k−3 for n≥k+2 and k≥3, and tp(BHn)=2n for n≥2. As corollaries, the pessimistic diagnosability of the known results about AGn and ANn is obtained directly.

Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosabilitytc(G) of G is the maximum number t for which G is conditionally t-diagnosable under the comparison model, while the 2-extra connectivityκ2(G) of a graph G is the minimum number k for which there is a vertex-cut F with |F|=k such that every component of G−F has at least 3 vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answers this problem by proving tc(G)=κ2(G) for a regular graph G with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, (n,k)-star graphs, alternating group networks, (n,k)-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, k-ary n-cube networks, dual-cubes, pancake graphs and hierarchical hypercubes as well. Furthermore, many known results about these networks are obtained directly.

The pessimistic diagnosabilities of some general regular graphs

The pessimistic diagnosis strategy is a classic strategy based on the PMC model. A system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by tp(G), is the maximal number of faulty processors so that the system G is t/t-diagnosable. In this paper, we study the pessimistic diagnosabilities of some general k-regular k-connected graphs Gn. The main result tp(Gn)=2k−2−g under some conditions is obtained, where g is the maximum number of common neighbors between any two adjacent vertices in Gn. As applications of the main result, every pessimistic diagnosability of many famous networks including some known results, such as the alternating group networks ANn, the k-ary n-cubes Qnk, the star graphs Sn, the matching composition networks G(G1,G2;M) and the alternating group graphs AGn, are obtained.

The Extra Connectivity and Conditional Diagnosability of Alternating Group Networks

Extra connectivity, diagnosability, and conditional diagnosability are all important measures for a multiprocessor system’s 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 $1\le h\le 3$ , and prove that the conditional diagnosability of an $n$ -dimensional alternating group graph, denoted by $AG_n$ , is $8n-27$ ( $n\ge 4$ ) under the PMC model. This is about four times of the $AG_n$ ’s traditional diagnosability. As a byproduct, the strong diagnosability of $AG_n$ is also obtained.

The number of shortest paths in the (n, k)-star graph

We count the shortest paths, not necessarily disjoint, between any two vertices in an (n, k)-star graph by counting the minimum factorizations of a permutation in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other structures closely related to the (n, k)-star graph, such as the alternating group networks. Furthermore, the techniques made use of in this paper can be applied to solve the same problem for some of the other structures such as the arrangement graph.

A note on the alternating group network

The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n,n−2)-star, a member of the well-known (n,k)-stars, 1≤k≤n−1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n−2.

Conditional diagnosability of alternating group networks

The growing size of a multiprocessor system increases its vulnerability to component failures. In order to maintain the system's 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.

Construction of vertex-disjoint paths in alternating group networks

The existence of parallel node-disjoint paths between any pair of nodes is a desirable property of interconnection networks, because such paths allow tolerance to node and/or link failures along some of the paths, without causing disconnection. Additionally, node-disjoint paths support high-throughput communication via the concurrent transmission of parts of a message. We characterize maximum-sized families of parallel paths between any two nodes of alternating group networks. More specifically, we establish that in a given alternating group network AN n , there exist n?1 parallel paths (the maximum possible, given the node degree of n?1) between any pair of nodes. Furthermore, we demonstrate that these parallel paths are optimal or near-optimal, in the sense of their lengths exceeding the internode distance by no more than four. We also show that the wide diameter of AN n is at most one unit greater than the known lower bound D+1, where D is the network diameter.

Internode Distance and Optimal Routing in a Class of Alternating Group Networks

Alternating group graphs AG <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , studied by Jwo and others, constitute a class of Cayley graphs that possess certain desirable properties compared with other regular networks considered by researchers in parallel and distributed computing. A different form, AN <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , of such graphs, proposed by Youhou and dubbed alternating group networks, has been shown to possess advantages over AG <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . For example, AN <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> has a node degree that is smaller by a factor of about 2 while maintaining a diameter comparable to that of AG <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , is maximally fault-tolerant, and shares some of the positive structural attributes of the well-known star graph. In this paper, we characterize the distance between any two nodes in AN <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and present an optimal (shortest-path) routing algorithm for this class of networks

Delay-period activity is affected by visual cues presented outside the memory field

In the wake of rapid development of multiprocessor systems, fault tolerance of processors plays an even more vital role in measuring the reliability of a multiprocessor system. The g-extra connectivity of a multiprocessor system modeled by graph G, denoted by κo(g)(G), is the minimum number of nodes whose deletion will disconnect the network and every remaining component has more than g vertices. The g-extra conditional diagnosability of multiprocessor system G, denoted by tg˜(G), is the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. Only a few achievements have been established on κo(g)(G) for some special graphs with small g. In this paper, we first determine that the g-extra connectivity of (n,k)-star network Sn,k is κo(g)(Sn,k)=n+g(k−2)−1 for 2≤k≤n−1 and 0≤g≤n−k and then show that the g-extra conditional diagnosability of Sn,k under the PMC model (n≥4, 2≤k≤n−1 and 1≤g≤n−k) and the MM* model (n≥6, 3≤k≤n−3 and 1≤g≤min⁡{n−k+14,k−2}) is tg˜(Sn,k)=n+g(k−1)−1, respectively. Meanwhile, we also show that Sn,k is (n+m(k−2)−1)/m-diagnosable under the pessimistic diagnostic strategy. As by-products, we derive the g-extra conditional diagnosability of n-dimensional star graph Sn (n≥4, g=1) and n-dimensional alternating group network ANn (n≥4, 1≤g≤2) under the PMC model.