Fault Tolerance Of Interconnection Networks Research Articles

The m-Component Connectivity of Leaf-Sort Graphs

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. As a special class of connectivity, m-component connectivity is a natural generalization of the traditional connectivity of graphs defined in terms of the minimum vertex cut. Moreover, it is a more advanced metric to assess the fault tolerance of a graph G. Let G=(V(G),E(G)) be a non-complete graph. A subset F(F⊆V(G)) is called an m-component cut of G, if G−F is disconnected and has at least m components (m≥2). The m-component connectivity of G, denoted by cκm(G), is the cardinality of the minimum m-component cut. Let CFn denote the n-dimensional leaf-sort graph. Since many structures do not exist in leaf-sort graphs, many of their properties have not been studied. In this paper, we show that cκ3(CFn)=3n−6 (n is odd) and cκ3(CFn)=3n−7 (n is even) for n≥3; cκ4(CFn)=9n−212 (n is odd) and cκ4(CFn)=9n−242 (n is even) for n≥4.

Star structure fault tolerance of Bicube networks

Processor and communication link failures are inevitable in a large multiprocessor system, and so the fault tolerance of its underlying interconnection network has become a key scientific issue. Connectivity is an important parameter to characterize network fault tolerance, and there are many novel variants of classical connectivity to measure the fault tolerance of interconnection networks. However, these new strategies only consider a single faulty vertex. Structure connectivity and substructure connectivity make up for this deficiency, which underline the fault situation with certain specific structures. H-structure-connectivity κ ( G ; H ) (resp. H-substructure-connectivity κ s ( G ; H ) ) of G is the minimum cardinality of H-structure-cuts (resp. H-substructure-cuts). For the n-dimensional Bicube network B Q n , we establish the structure and substructure connectivity of Bicube networks, i.e. κ ( B Q n ; K 1 , 1 ) = κ s ( B Q n ; K 1 , 1 ) = n for odd n ≥ 5 ; κ ( B Q n ; K 1 , 1 ) = κ s ( B Q n ; K 1 , 1 ) = n − 1 for even n ≥ 4 and κ ( B Q n ; K 1 , r ) = κ s ( B Q n ; K 1 , r ) = ⌈ n 2 ⌉ for n ≥ 6 and 2 ≤ r ≤ n − 1 .

Many-to-many edge-disjoint paths in (n,k)-enhanced hypercube under three link-faulty hypotheses

One of the most central issues in interconnection networks of parallel and distributed systems is finding edge-disjoint paths that transmit information. Finding as many as possible many-to-many edge-disjoint paths is conducive to improving the fault-tolerance of such networks. As an interconnection network topology, (n,k)-enhanced hypercube Qn,k (1≤k≤n−1), is a momentous variant of well-known hypercube. For integers 1≤l≤n−1 and n≥2, let δ=0 if 1≤l≤n−k, and δ=1 if n−k+1≤l≤n−1. This paper offers a unified method to determine the minimum cardinalities of faulty links in Qn,k, whose malfunction divides this network into several connected components such that each processor has at least l+δ neighbors, each component contains no less than 2l processors and the number of average neighbors for all processors is at least l+δ, respectively. Under these three different link-faulty assumptions, but for the condition of k=2 and l=n−2, the minimum cardinalities of such faulty links share the same value (n−l−δ+1)2l. And the value in the exceptional case is (n−l−δ)2l+1=2n−1. In other words, we find the maximum numbers of many-to-many edge-disjoint paths of Qn,k under the above three hypotheses, which offers refined measurements for the reliability and fault-tolerance of interconnection networks.

The generalized 4-connectivity of folded hypercube

ABSTRACT In recent years, a growing number of concepts have been proposed to better assess network fault tolerance, among which the generalized -connectivity has been widely used in the research of fault tolerance of interconnection networks. For connected graph , refers to the maximum number of internally disjoint trees in to connect , where with . The generalized -connectivity of G . The -dimensional folded hypercube , as a hypercube-like network, is obtained by adding edges on -dimensional hypercube . In this paper, we discuss the generalized 4-connectivity of and show that for .

The Sufficient Conditions of (δ(G) − 2)-(|F|-)Fault-Tolerant Maximal Local-(Edge-)Connectivity of Connected Graphs

An interconnection network is usually modeled by a connected graph in which vertices represent processors and edges represent links between processors. The connectivity is an important parameter to evaluate the fault tolerance of interconnection networks. A connected graph [Formula: see text] is maximally local-(edge-)connected if each pair vertices [Formula: see text] of [Formula: see text] is connected by min[Formula: see text] pairwise (edge-)disjoint paths between [Formula: see text] and [Formula: see text] in [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] ([Formula: see text]) with [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text] if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] with [Formula: see text], where [Formula: see text] is a conditional faulty vertex (edge) set of order [Formula: see text]. In this paper, we obtain the sufficient condition of connected graphs to be [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Moreover, we consider the sufficient condition of connected graphs to be [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text]. Some previous results in [Theor. Comput. Sci. 731 (2018) 50–67] and [Theor. Comput. Sci. 847 (2020) 39–48] are extended.

Fault-Tolerant Maximal Local-Edge-Connectivity of Augmented Cubes

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A connected graph [Formula: see text] is said to be maximally local-edge-connected if each pair of vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are connected by [Formula: see text] pairwise edge-disjoint paths. In this paper, we show that the [Formula: see text]-dimensional augmented cube [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; under the restricted condition that each vertex has at least three fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; and under the restricted condition that each vertex has at least [Formula: see text] fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Furthermore, we show that a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1, a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1.

The preclusion numbers and edge preclusion numbers in a class of Cayley graphs

Let G be a hierarchical network (graph) with vertex set V(G) and edge set E(G). The preclusion set of the subnetwork G′ (defined as a smaller network but with the same topological properties as the original one) in G is a subset V′ of V(G) such that G−V′ has no any subnetwork isomorphic to G′. The preclusion number of G′ in G is F(G′)=min{|V′|:V′ is the preclusion set of G′}. Similarly, the edge preclusion set of G′ in G is a subset E′ of E(G) such that G−E′ has no any subnetwork isomorphic to G′. The edge preclusion number of G′ in G is f(G′)=min{|E′|:E′ is the edge preclusion set of G′}. The preclusion number and edge preclusion number are parameters which measure the fault tolerance of interconnection networks in the presence of failures. In this paper, we investigate the two parameters for a class of Cayley graphs T2n+1 generated by the transposition tree similar to the star. Let m be an integer with 0≤m≤n−1, let F(n,m)=F(T2(n−m)+1), and let f(n,m)=f(T2(n−m)+1). We prove that F(n,0)=f(n,0)=1, F(n,1)=2n(2n+1) with 2n+1 prime, f(n,1)≤4n(2n+1) with 2n+1 prime, F(n,n−1)=(2n+1)!6, f(n,n−1)=n(2n+1)!6, and 2n+12m(2m)!≤F(n,m)≤f(n,m)≤nm2n+12m(2m)!. Finally, we prove that F(n,m) is of order at most n3m−1.

Fault-Tolerant Maximal Local-Connectivity on Cayley Graphs Generated by Transpositions

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.

A Note on the Connectivity of m-Ary n-Dimensional Hypercubes

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. As a topology structure of interconnection networks, the m-ary n-dimensional hypercube [Formula: see text] has many good properties. In this paper, we prove, by elementary method, that [Formula: see text] is tightly [Formula: see text] super connected [Formula: see text] and super edge-connected [Formula: see text].

The extra connectivity of the enhanced hypercubes

As one of the best parameters to evaluate the fault tolerance of interconnection networks, extra connectivity plays an important role in multiprocessor systems for reliable computing. Given a graph G and a non-negative integer g, the g-extra connectivity (resp. g-extra edge connectivity) of G, denoted by κg(G) (resp. λg(G)), is the minimum cardinality of a set of vertices (resp. edges) in G, if exists, whose deletion disconnects G and leaves each remaining component with at least g+1 vertices. In this paper, we determine κg(Qn,k) and λg(Qn,k) of the enhanced hypercube Qn,k for g=1,2. Furthermore, as the byproduct of our results, we determine the conditional diagnosability of Qn,k under the comparison model and PMC model.

Diagnosability of arrangement graphs with missing edges under the MM* model

Diagnosability is an important parameter to measure the fault tolerance of interconnection networks. Arrangement graph is a generalisation of the star graphs, yet it is more flexible in its size than the star graphs. In this paper, we study the local diagnosability of , and show that it has the strong local diagnosability property even if there exist missing edges in it under the MM* model, and the result is optimal with respect to the number of missing edges.

Edge fault tolerance of interconnection networks with respect to maximally edge-connectivity

The interconnection networks are generally modeled by a connected graph G, and the connectivity of G is an important parameter for reliability and fault tolerance of the network. A connected graph G=(V,E) is maximally edge-connected (or maximally-λ for short) if its edge-connectivity attains its minimum degree. We define a maximally-λ graph G to be m-maximally-λ if G−S is still maximally-λ for any edge subset S⊆E(G) with |S|≤m. The maximum integer of such m, denoted by mλ(G), is said to be the edge fault tolerance of G with respect to the maximally-λ property. In this paper, we discuss the edge fault tolerance for maximally-λ property of two families of interconnection networks, from which we determine the exact values of mλ(G) for some well-known networks.

The connectivity and nature diagnosability of expandedk-aryn-cubes

Connectivity and Diagnosability play an important role in measuring the fault tolerance of interconnection networks. As a topology structure of interconnection networks, the expanded k -ary n -cube X Q k n has many good properties. In this paper, we prove that (1) the connectivity of X Q k n is 4 n ; (2) the nature connectivity of X Q k n is 8 n − 4; (3) the nature diagnosability of X Q k n under the PMC model and MM ∗ model is 8 n − 3 for n ≥ 2.

On the reliability of generalized Petersen graphs

The super-connectivity (super-edge-connectivity) of a connected graph G is the minimum number of vertices (edges) that need to be deleted from G in order to disconnect G without creating isolated vertices. We determine when the generalized Petersen graphs GP[n,k] are super-connected and super edge-connected, and show that their super-connectivity and their super-edge-connectivity are both equal to four when n∉{2k,3k}. These results partially answer a question by Harary (1983) and are of interest especially in the study of reliability and fault tolerance of interconnection networks, since the graphs in this class are good candidates for such networks.

The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. The g-good-neighbor connectivity of an interconnection network G is the minimum cardinality of g-good-neighbor cuts. Diagnosability of a multiprocessor system is one important study topic. A new measure for fault diagnosis of the system restrains that every fault-free node has at least g fault-free neighbor vertices, which is called the g-good-neighbor diagnosability of the system. As a famous topology structure of interconnection networks, the n-dimensional bubble-sort star graph BSn has many good properties. In this paper, we prove that 2-good-neighbor connectivity of BSn is 8n−22 for n≥5 and the 2-good-neighbor connectivity of BS4 is 8; the 2-good-neighbor diagnosability of BSn is 8n−19 under the PMC model and MM∗ model for n≥5.

A Family of Fault-Tolerant Efficient Indirect Topologies

On the one hand, performance and fault-tolerance of interconnection networks are key design issues for high performance computing (HPC) systems. On the other hand, cost should be also considered. Indirect topologies are often chosen in the design of HPC systems. Among them, the most commonly used topology is the fat-tree. In this work, we focus on getting the maximum benefits from the network resources by designing a simple indirect topology with very good performance and fault-tolerance properties, while keeping the hardware cost as low as possible. To do that, we propose some extensions to the fat-tree topology to take full advantage of the hardware resources consumed by the topology. In particular, we propose three new topologies with different properties in terms of cost, performance and fault-tolerance. All of them are able to achieve a similar or better performance results than the fat-tree, providing also a good level of fault-tolerance and, contrary to most of the available topologies, these proposals are able to tolerate also faults in the links that connect to end nodes.

Hamiltonian Embedding in Crossed Cubes with Failed Links

The crossed cube is a prominent variant of the well known, highly regular-structured hypercube. In [24], it is shown that due to the loss of regularity in link topology, generating Hamiltonian cycles, even in a healthy crossed cube, is a more complicated procedure than in the hypercube, and fewer Hamiltonian cycles can be generated in the crossed cube. Because of the importance of fault-tolerance in interconnection networks, in this paper, we treat the problem of embedding Hamiltonian cycles into a crossed cube with failed links. We establish a relationship between the faulty link distribution and the crossed cube's tolerability. A succinct algorithm is proposed to find a Hamiltonian cycle in a CQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> tolerating up to n-2 failed links.

The Conditional Connectivity of (&lt;i&gt;n,k&lt;/i&gt;)-Star Graph

Conditional connectivity has been proposed as an important parameter to estimate the fault tolerance of interconnection networks. In this paper, we consider the conditional connectivity of (n, k)-star graph. An (n, k)-star graph with dimension n (n ≥ 4) and order k can be partitioned into n subgraphs,. By utilizing this property, we give and proof the minimal cut-set and the minimal conditional cut-set of S n,k. We hence obtain that the conditional connectivity of (n, k)-star graph S n,k is n+k−3.

An analysis of edge fault tolerance in recursively decomposable regular networks

Fault tolerance of interconnection networks is one of the major considerations in evaluating the reliability of large scale multiprocessor systems. In the paper, the reliability of a family of regular networks with respect to edge failures is investigated using four different fault tolerance measures. Two probabilistic measures, resilience and restricted resilience, are developed, used to evaluate disconnection likelihoods using two different failure models, and compared with corresponding deterministic measures. The network topologies chosen for the present study all have the recursive decomposition property, where larger networks can be decomposed into copies of smaller networks of the same topology. This family of graphs includes the k-ary n-cube, star and cube connected cycle graphs, which have optimal deterministic connectivities. The probabilistic fault tolerance measures, however, are found to depend on topological properties such as network size and degree. >

Fault tolerance analysis of the class of rearrangeable interconnection networks

The issue of the fault tolerance of interconnection networks has received considerable attention since the last decade. The theoretical framework developed by J. P. Shen and J. P. Hays ( IEEE Trans. Comput. C-33, 3 (Mar. 1984), 241–248) and extended by S. Pakzad and S. Lakshmivarahan ( Internat. J. Comput. Syst. Sci. Engrg. 3, 2 (April 1988), 91–99) can be used as a guideline to analyze the fault tolerance behavior of the interconnection networks in a multicomputer environment. This paper analyzes the fault tolerance characteristics of a range of rearrangeable β-networks based on the concepts and the framework developed by S. Pakzad and S. Lakshmivarahan (cited earlier). These rearrangeable β-networks include the Benes network, the Waksman network, the Joel network, and the serial network. In addition, this paper presents a comparative analysis of the aforementioned networks according to their hardware cost, performance, and degree of fault tolerance.