Faulty Vertices Research Articles

Cycles Embedding in Enhanced Hypercubes with Faulty Vertices

Cycles Embedding in Enhanced Hypercubes with Faulty Vertices

Reliability of the round matching composition networks based on g-extra conditional fault

Connectivity and diagnosability are very important parameters for the tolerant of an interconnection network. Let F be a vertex set of graph G, the g-extra connectivity k˜(g)(G) of G requires each component of G−F has at least (g+1) vertices. The g-extra conditional diagnosability t˜(g)(G) is the maximum number of faulty vertices that the graph can be guaranteed to identify under the condition that each fault-free component has at least g+1 vertices. In this paper, we study the problem about the g-extra connectivity k˜(g)(G) of the round matching composition networks, and obtain some results about the g-extra conditional diagnosability of the round matching composition networks under the PMC model and MM* model, respectively.

The 3-Extra Connectivity of the Data Center Network BCube

Abstract Connectivity is a significant metric to assess the fault tolerance of a network. For a faulty vertex set $H$, the $h$-extra connectivity is defined under the assumption that every component of the network removing $H$ has at least $h+1$ fault-free vertices. Compared to the traditional connectivity, which is defined under the assumption that the network removing $H$ is disconnected or trivial, the $h$-extra connectivity can better reflect the true fault tolerance of the network. The $BCube$ is an important server-centric data center network; it has good fault tolerance and scalability. In this paper, our research focuses on the logical structure of $BCube$, named $BC_{n,k}$, which is actually a specific type of generalized hypercubes. We prove that the 3-extra connectivity of $BC_{n,k}$ is $4(k+1)(n-1)-4n$ for $k\geq 4$ and $n\geq 4$.

Symmetric PMC model of diagnosis, b-matchings in graphs and fault identification in t-diagnosable systems

A major breakthrough in the diagnosis of t-diagnosable systems occurred when Dahbura and Mason developed a diagnosis algorithm under the PMC model using the matching theory in bipartite graphs. In this paper we introduce a new testing model called the symmetric PMC(SPMC) model. Our main contributions are as follows: We prove that the diagnosability of an n-dimensional hypercube under the SPMC model is almost twice its diagnosability under the PMC model. We then show that the fault diagnosis problem for a t-diagnosable system under the SPMC model reduces to that of determining a maximum weighted b-matching in its diagnosis graph. Algorithm LABEL is then given to identify all the faulty vertices using the maximum b-matching of the diagnosis graph. Finally, using certain results from the theory of partitions of integers we establish the worst case complexity of our t-diagnosis algorithm. The complexity is much better than the worst case complexity of the Dahbura-Mason algorithm for the t-diagnosable problem under the PMC model.

A Complete Fault Tolerant Method for Extra Fault Diagnosability of Alternating Group Graphs

A network's diagnosability is the maximum number of faulty vertices that the network can discriminate solely by performing mutual tests among vertices. The original diagnosability without any condition is often rather low because it is bounded by the network's minimum degree. The h-extra fault diagnosability is an important and widely accepted diagnostic strategy as a new measure of diagnosability, which guarantees that the scale of every component is at least h+1 in the remaining system. Moreover, it increases the allowed faulty vertices, hence enhancing the diagnosability of the network. There have been lots of state-of-the-art literatures concerning the h-extra fault diagnosability. Although there are some methods to theoretically prove the extra fault diagnosability of some other well-known networks under MM* model, these methods have some serious flaws when there exists a 4-cycle in these networks. In this article, we investigate the reason that caused the flawed results in some references, and we derive a different, broadly applicable, and complete fault tolerant method to establish the extra fault diagnosability in an n-dimensional alternating group graph AG <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> under MM* model. The complete fault tolerant method adopts combinatorial properties and linearly many fault analysis to conquer the core of our proofs. Moreover, we compare the extra fault diagnosability of AG <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> with various types of fault diagnosability, including the diagnosability, strong diagnosability, conditional diagnosability, t/k-diagnosability, and pessimistic diagnosability. It can be seen that the extra fault diagnosability is greater than all the other types of fault diagnosability.

Component conditional fault tolerance of hierarchical folded cubic networks

For the sake of achieving higher reliability, conditional connectivity has gradually become well-known. Component connectivity, as a kind of conditional connectivity, is an extension of the classical connectivity. Given a simple and undirected graph G and a nonnegative integer r, the (r+1)-component connectivity of graph G, say cκr+1(G), is the minimum number of a vertex cut whose removal causes the surviving graph to have at least r+1 components. Another basic parameter of reliability, diagnosability is often related to the number of components in the surviving graph. The (r+1)-component diagnosability, say ctr+1(G), is the maximum size of fault sets subject to at least r+1 components in the surviving graph, provided that all faulty vertices can be detected. A graph is t/k-diagnosable if all faulty vertices can be isolated into a set in which up to k vertices are fault-free, provided that the number of faulty vertices is at most t. As a two-level interconnection network, the hierarchical folded cubic network HFQ(n) possesses a great deal of nice properties. In this paper, we first show that the n-dimensional hierarchical folded cubic network is tightly super connected. Then, we explore that cκr+1(HFQ(n))=r(n+1)−(r2)+1(n≥4,1≤r≤n−4), and obtain that ctr+1(HFQ(n))=(r+1)n−(r2)+2(n≥4,1≤r≤n−4) under the PMC and MM* models. Last, we show that the hierarchical folded cubic network HFQ(n) is [(k+1)n−(k2)+2]/k-diagnosable, where n≥4 and 1≤k≤n−4.

The Reliability ofk-Aryn-Cube Based on Component Connectivity

AbstractConnectivity and diagnosability are two crucial subjects for a network’s ability to tolerate and diagnose faulty processors. The $r$-component connectivity $c\kappa _{r}(G)$ of a network $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. The $r$-component diagnosability $ct_{r}(G)$ of a network $G$ is the maximum number of faulty vertices that the system can guarantee to identify under the condition that there exist at least $r$ fault-free components. This paper first establishes that the $(r+1)$-component connectivity of $k$-ary $n$-cube $Q^{k}_{n}$ is $c\kappa _{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{1}{2}\Big)r+1$ for $n\geq 2$, $k\geq 4$ and $1\leq r\leq n$. In view of $c\kappa _{r+1}(Q^{k}_{n})$, we prove that the $(r+1)$-component diagnosabilities of $k$-ary $n$-cube $Q^{k}_{n}$ under the PMC model and MM* model are $ct_{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{3}{2}\Big)r+2n$ for $n\geq 4$, $k\geq 4$ and $1\leq r\leq n-1$.

The Relationship Between theg-Extra Connectivity and theg-Extra Diagnosability of Networks Under the MM* Model

AbstractMotivated by $g$-extra connectivity, the $g$-extra diagnosability is proposed as a better and more realistic measurement for fault diagnosis of interconnection networks, which is defined as the maximum number of faulty vertices that can be identified when each remaining component has no fewer $g+1$ vertices. Under the MM* model, a variety of interconnection networks’ $g$-extra diagnosability have been investigated, such as hypercube, folded hypercube, $(n,k)$-star network, alternating group graph, etc. These results mostly share similar derivation processes to derive the $g$-extra diagnosability of involved networks by using the $g$-extra connectivity. Therefore, a general approach to derive the $g$-extra diagnosability of a network from its $g$-extra connectivity was investigated in (Wang, S. Y. and Wang, M. (2019) The $g$-good-neighbor and $g$-extra diagnosability of networks. Theor. Comput. Sci., 773, 107–114) and (Huang, Y. Z., Lin, L. M. and Xu, L. (2020) A new proof for exact relationship between extra connectivity and extra diagnosability of regular connected graphs under MM* model. Theor. Comput. Sci., 828–829, 70–80). However, there are some shortcomings in both references. By summarizing the existing shared practices, we propose a new relationship between the $g$-extra connectivity and the $g$-extra diagnosability of networks under the MM* model. As applications, we derive the $g$-extra diagnosability of bijective connection networks and $(n,k)$-star graphs.

Reliability Evaluation of Generalized Exchanged Hypercubes Based on Imprecise Diagnosis Strategies

Fault diagnostic analysis is extremely important for interconnection networks. The [Formula: see text]-diagnosis imprecise strategy plays an essential role in the reliability of networks. The [Formula: see text]-diagnosis strategy can detect up to [Formula: see text] faulty vertices which might include at most [Formula: see text] misdiagnosed vertices. The exchanged hypercube is obtained by systematically removing links from a binary hypercube, which has smaller maximum degree and Wiener index than the hypercube. We use [Formula: see text] to denote the generalized exchanged hypercube, and show in this paper that [Formula: see text] is [Formula: see text]-diagnosable with [Formula: see text] and [Formula: see text] under the PMC model and MM[Formula: see text] model. We also propose a [Formula: see text]-diagnosis algorithm on [Formula: see text]. As a side benefit, the [Formula: see text]-diagnosability of the dual-cube-like network [Formula: see text] can be directly obtained from our results.

Fault-tolerant hamiltonian cycles and paths embedding into locally exchanged twisted cubes

The foundation of information society is computer interconnection network, and the key of information exchange is communication algorithm. Finding interconnection networks with simple routing algorithm and high fault-tolerant performance is the premise of realizing various communication algorithms and protocols. Nowadays, people can build complex interconnection networks by using very large scale integration (VLSI) technology. Locally exchanged twisted cubes, denoted by (s + t + 1)-dimensional LeTQs,t, which combines the merits of the exchanged hypercube and the locally twisted cube. It has been proved that the LeTQs,t has many excellent properties for interconnection networks, such as fewer edges, lower overhead and smaller diameter. Embeddability is an important indicator to measure the performance of interconnection networks. We mainly study the fault tolerant Hamiltonian properties of a faulty locally exchanged twisted cube, LeTQs,t − (fv + fe), with faulty vertices fv and faulty edges fe. Firstly, we prove that an LeTQs,t can tolerate up to s − 1 faulty vertices and edges when embedding a Hamiltonian cycle, for s ⩾ 2, t ⩾ 3, and s ⩽ t. Furthermore, we also prove another result that there is a Hamiltonian path between any two distinct fault-free vertices in a faulty LeTQs,t with up to (s − 2) faulty vertices and edges. That is, we show that LeTQs,t is (s − 1)-Hamiltonian and (s − 2)-Hamiltonian-connected. The results are proved to be optimal in this paper with at most (s − 1)-fault-tolerant Hamiltonicity and (s − 2) fault-tolerant Hamiltonian connectivity of LeTQs,t.

Every edge lies on cycles of folded hypercubes with a pair of faulty adjacent vertices

An n-dimensional folded hypercube FQn is a well-known extended structure of an n-dimensional hypercube Qn, which can be constructed from Qn by adding an edge to every pair of vertices with complementary addresses. FQn possesses many properties that are superior to those of Qn including the diameter, fault diameter, and connectivity. It has been shown that FQn is bipartite for every odd n≥3, and non-bipartite for every even n≥2. In this report, we let {f1,f2} denote the set of a pair of adjacent faulty vertices in FQn. We then show the cycle embedding properties as follows:1. For n≥3, every fault-free edge can lie on a cycle of any even length l with 4≤l≤2n−4 in FQn−{f1,f2};2. For every even n≥4, every fault-free edge can lie on a cycle of every odd length l with n+1≤l≤2n−3 in FQn−{f1,f2}.Our study improves the result of the previous study with respect to the embedding of the longest cycle in a non-bipartite FQn, and the quantity of the faulty vertices in FQ3.

(n − 2)-Fault-Tolerant Edge-Pancyclicity of Crossed Cubes CQn

As one of the most fundamental networks for parallel and distributed computation, cycle is suitable for developing simple algorithms with low communication cost. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant edge-pancyclic if after deleting any faulty set [Formula: see text] of [Formula: see text] vertices and/or edges from [Formula: see text], every correct edge in the resulting graph lies in a cycle of every length from [Formula: see text] to [Formula: see text], inclusively, where [Formula: see text] is the girth of [Formula: see text], the length of a shortest cycle in [Formula: see text]. The [Formula: see text]-dimensional crossed cube [Formula: see text] is an important variant of the hypercube [Formula: see text], which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of [Formula: see text], and shows that if [Formula: see text] contains at most [Formula: see text] faulty vertices and/or edges then, for any fault-free edge [Formula: see text] and every length [Formula: see text] from [Formula: see text] to [Formula: see text] except [Formula: see text], there is a fault-free cycle of length [Formula: see text] containing the edge [Formula: see text]. The result is optimal in some senses.

The bipanconnectivity of bipartite hypercube-like networks

Bipanconnectivity is an important parameter in bipartite networks related on the embedding problem of linear arrays and rings. In this paper, we study the fault-tolerant bipanconnectivity of bipartite n-dimensional hypercube-like networks, denoted as Bn′. We show that for any n-dimensional bipartite hypercube-like network G∈Bn′ with f faulty elements (edges and/or vertices), including fv faulty vertices such that f≤n−2, for each pair of fault-free vertices of distance d in G, there exists a fault-free path of length l linking them, where 2n−4≤l≤2n−2fv−1 and l−d≡0 (mod 2). Our result is optimal when considering the number of faulty elements.

Reliability of DQcube Based on g-Extra Conditional Fault

Abstract Diagnosability and connectivity are important metrics for the reliability and fault diagnosis capability of interconnection networks, respectively. The g-extra connectivity of a graph G, denoted by $\kappa _g(G)$, is the minimum number of vertices whose deletion will disconnect the network and every remaining component has more than $g$ vertices. The g-extra conditional diagnosability of graph G, denoted by $t_g(G)$, is the maximum number of faulty vertices that the graph G can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. In this paper, we first determine that g-extra connectivity of DQcube is $\kappa _g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}$ for $0\leq g\leq n-3$ and then show that the g-extra conditional diagnosability of DQcube under the PMC model $(n\geq 4, 1\leq g\leq n-3)$ and the MM$^\ast$ model $(n\geq 7, 1\leq g\leq \frac{n-3}{4})$ is $t_g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}+g$, respectively.

On g-Extra Conditional Diagnosability of Twisted Hypercubes under MM∗ Model

Connectivity and diagnosability are important parameters in measuring the reliability and fault-tolerance of an interconnection network [Formula: see text]. The [Formula: see text]-extra conditional faulty set [Formula: see text] is a faulty vertex set such that every component of [Formula: see text] has at least [Formula: see text] vertices. The [Formula: see text]-extra connectivity [Formula: see text] of a connected graph [Formula: see text] is the minimum cardinality of a [Formula: see text]-extra conditional faulty set [Formula: see text] of [Formula: see text] such that [Formula: see text] is disconnected. The [Formula: see text]-extra conditional diagnosability [Formula: see text] of a graph [Formula: see text] is the maximum value of [Formula: see text] such that [Formula: see text] is [Formula: see text]-extra conditionally [Formula: see text]-diagnosable. The [Formula: see text]-extra connectivity of [Formula: see text] is necessary for [Formula: see text]-extra diagnosability of [Formula: see text]. The [Formula: see text]-dimensional twisted hypercube [Formula: see text] is a new variant of hypercubes with asymptotically optimal diameter. In this paper, we first give the [Formula: see text]-extra connectivity of [Formula: see text] for [Formula: see text] and [Formula: see text]; and then obtain the [Formula: see text]-extra conditional diagnosability of [Formula: see text] for [Formula: see text] and [Formula: see text] under the MM[Formula: see text] model.

The g-extra diagnosability of the generalized exchanged hypercube

Diagnosability of a self-diagnosable interconnection structure specifies the maximum number of faulty vertices such a structure can identify by itself. A variety of diagnosability models have been suggested. It turns out that a diagnosability property of a network structure is closely associated with its relevant connectivity property. Based on this observation, a general diagnosability derivation process has been suggested. The g-extra connectivity of a graph G characterizes the size of a minimum vertex set F such that, when it is removed, every component in the disconnected survival graph, contains at least g + 1 vertices. In this paper, we discuss the aforementioned general derivation process, derive the g-extra connectivity, and then apply the aforementioned general process to reveal the g-extra diagnosability of the generalized exchanged hypercube.

A Novel Low Cost Interconnection Architecture Based on the Generalized Hypercube

The generalized hypercube (GH) is one key interconnection network with excellent topological properties. It contains many other interconnection topologies, such as the hypercube network, the complete graph, the mesh network, and the $k$ k -ary $n$ n -cube network. It can also be used to construct some data center networks, such as HyperX, BCube, FBFLY, and SWCube. However, the construction cost of GH is high since it contains too many links. In this paper, we propose a novel low cost interconnection architecture called the exchanged generalized hypercube (EGH). We study the properties of EGH, such as the number of edges, the degree of vertices, connectivity, diameter, and diagnosability. Then, we give a routing algorithm to find the shortest path between any two distinct vertices of EGH. Furthermore, we design an algorithm to give disjoint paths between any two distinct vertices of EGH. In addition, we propose two local diagnosis algorithms: LDT $_{EGH}$ E G H and LDWB $_{EGH}$ E G H in EGH under PMC model and MM model, respectively. Simulation results demonstrate that even if the proportion of faulty vertices in EGH is up to 25 percent, the probability that these two diagnosis algorithms can successfully determine the status of vertices is more than 90 percent. As far as the number of edges is concerned, the analysis shows that the construction cost of EGH is much less than that of GH. We could regard this work as the basis for proposing future new high performance topologies.

Rg conditional diagnosability: A novel generalized measure of system-level diagnosis

System-level diagnosis has become an important diagnosis method for multiprocessor systems. Among all system-level diagnosis measures, diagnosability is relatively small. The conditional diagnosability constraint that each vertex has at least one good neighbor is relatively conservative when the dimension is far greater than 1, and g-good-neighbor conditional diagnosability does not consider this restriction on faulty vertices. Therefore, a thorough study of diagnosability under the condition that each vertex has at least g good neighbors is an appealing subject. Motivated by Rg vertex connectivity, in this paper, we introduce a novel generalized system-level diagnosis measure named Rg conditional diagnosability, which assumes that every processor has at least g good neighbors. The popular conditional diagnosability is a special case of Rg conditional diagnosability when g=1. Then, we determine that the Rg conditional diagnosability of n-dimensional hypercube Qn under the Preparata Metze Chien (PMC) model is 22g(n−2g)+22g−1−1.

Reliability analysis of data center networks based on precise and imprecise diagnosis strategies

Fault tolerance and reliability are the crucial issues for data center networks (DCNs). Both the g-extra conditional diagnosis (g-ECD) precise strategy and the t/k-diagnosis imprecise strategy play essential role in the reliability of networks. For the data center network based on DCell structure Dm,n, we prove that: 1) the g-ECD of Dm,n under the MM⁎ model is (g+1)m+n−1 with n≥4, m≥3, and 1≤g≤min⁡{n4,m−2}; 2) Dm,n is [(k+1)(m−1)+n]/k-diagnosable under the MM⁎ model with n≥2, m≥2, and 0≤k≤n−1. Furthermore, for N-server DCN, we propose the first t/k-diagnosis algorithm on Dm,n under the MM⁎ model, namely t/k-Dm,n-DIAG with O(NlogN) time complexity. Comparing with the traditional t-diagnosable algorithm by Ziwich and Duarte (2016) [11], on Dm,n, the t/k-Dm,n-DIAG can identify the number of faulty vertices is almost k times larger than the t-diagnosable algorithm, where the time complexity of t-diagnosable algorithm is about O(N(logN)3). These results provide a quantitative analysis for the reliability and availability evaluation of a large-scale DCN.

Fault-Tolerant Panconnectivity of Augmented Cubes AQn

The augmented cube [Formula: see text] is a variation of the hypercube [Formula: see text]. This paper considers the fault-tolerant Panconnectivity of [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. We prove that for any two fault-free vertices [Formula: see text] and [Formula: see text] with distance [Formula: see text] in [Formula: see text], there exists a fault-free path [Formula: see text] of each length from [Formula: see text] to [Formula: see text] in [Formula: see text] if [Formula: see text], where [Formula: see text] is the number of faulty vertices in [Formula: see text]. Moreover, the bound is sharp.