Well-known Interconnection Networks Research Articles

Characterization of matroidal connectivity of regular networks

High performance computing system, which takes an interconnection network as its infrastructure topology, has been utilized in scientific computing, big data analysis as well as artificial intelligence. With the rapid growth of infrastructure topology (interconnection network) in high performance computing system, the probability of network element malfunction increases dramatically. Generally, the reliability of interconnection network is measured by the traditional (edge) connectivity, which is limited by the minimum degree of the network. In order to overcome the defect, the matroidal connectivity has been newly proposed to extend the edge connectivity. For the regular network G, let Bk(G)={B1,B2,…,Bk} be a fixed edge partition ordered sequence of G and χk=(x1,x2,…,xk) be a sequence of k integers. For any D⊆E(G), if |D∩Bi|≤xi(i∈[k]) and G−D is disconnected, then D is named matroidal cut set of G. The matroidal connectivity of G, denoted by λ(G,Bk(G)), is the minimum size of matroidal cut set of G. In this work, we characterize the matroidal connectivity of a class of n-dimensional regular networks Gn. To be more specific, we show that λ(Gn,Bn(Gn))=(k−l+1)f(l+j), where f(n) is a function about n, and l,k,n are positive integers (l≥1), j=n−k. As empirical analysis, we directly determine the matroidal connectivity of some well-known interconnection networks, such as hypercube Qn, star graph STn and alternating group graph AGn.

Component Connectivity of Alternating Group Networks and Godan Graphs

Connectivity is an important index to evaluate the reliability and fault tolerance of a graph. As a natural extension of the connectivity of graphs, the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components. It is a scientific issue to determine the exact values of [Formula: see text] for distinguishing the fault tolerability of networks. However, [Formula: see text]-component connectivity of many well-known interconnection networks has not been explored even for small [Formula: see text]. For the [Formula: see text]-dimensional alternating group networks [Formula: see text] and [Formula: see text]-dimensional godan graphs [Formula: see text], we show that [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text] and [Formula: see text].

Reliability and conditional diagnosability of hyper bijective connection networks

The g-extra connectivity and diagonalisability are two important metrics to fault-tolerance and robustness of a multiprocessor system whose network structure is modelled by a graph. In this work, we explore the reliability of a newly proposed network called hyper bijective connection networks (HBC, for short), which is an extension of the family of the well-known interconnection networks, such as hypercube and its variants. We prove that 2-extra vertex connectivity and 3-extra vertex connectivity of n-dimensional HBC are 3n + m−6 for and and 4n + m−8 for and , respectively. Using its desirable fault-tolerance, we show that the conditional diagonalizabilities of n-dimensional HBC under the PMC model are m + 4n−7 (resp., 4n−5) for and (resp., m = 3 and ) and its conditional diagonalizability under MM model is m + 3n−6 for and .

{1,2,3}-Restricted Connectivity of $(n,k)$-Enhanced Hypercubes

Abstract The connectivity of a graph is a classic measure for fault tolerance of the network. Restricted connectivity measure is a crucial subject for a multiprocessor system’s ability to tolerate fault processors, and improves the connectivity measurement accuracy. Furthermore, if a network possesses a restricted connectivity property, it is more reliable with a lower vertex failure rate compared with other networks. The $\left (n,k\right )$-dimensional enhanced hypercube, denoted by $Q_{n,k}$, a variant of hypercube, which is a well-known interconnection network. In this paper, we analyze the fault tolerant properties for $\left (n,k\right )$-enhanced hypercube, and establish the $1$-restricted connectivity of $Q_{n,k} (n\ge k+1)$ and $\{2,3\}$-restricted connectivity of $(n,k)$-enhanced hypercube $Q_{n,k} (n=k+1)$. Furthermore, we propose the tight upper bound of $\{2,3\}$-restricted connectivity of $Q_{n,k} (n> k+1)$. Moreover, we show many figures to better illustrate the process of the proofs.

The extra connectivity of enhanced hypercubes

ABSTRACTConnectivity plays an crucial role in measuring the fault tolerance of a multiprocessor system in the case of vertices failures. Extra connectivity is an important indicator of a network's ability for diagnosis and fault tolerance. In this paper, we analyse the fault tolerance ability of the -dimensional enhanced hypercube, a well-known interconnection network denoted by proposed for multiprocessor systems, and establish the -extra connectivity of .

The h-extra connectivity of k-ary n-cubes

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The h-extra connectivity κh(G) is an important subject for a multiprocessor system's reliability to tolerate faulty processors. It is defined as the minimum cardinality of a set of vertices in G, whose deletion disconnects G and leaves every remaining component with more than h vertices. In this paper, we investigate the h-extra connectivity of k-ary n-cube Qnk, a well-known interconnection network proposed for multiprocessor systems, and showκh(Qnk)={(h+1)2n−2h−(h2), if 0≤h≤n,n≥2,k≥4;(h+1)2n−3h−(h2), if 0≤h≤n,n≥3,k=3, except κ2(Q24)=6.

Minimum Neighborhood of Alternating Group Graphs

The minimum neighborhood and combinatorial property are two important indicators of fault tolerance of a multiprocessor system. Given a graph $G$ , $\theta _{G}(q)$ is the minimum number of vertices adjacent to a set of $q$ vertices of $G$ ( $1\leq q\leq |V(G)|$ ). It is meant to determine $\theta _{G}(q)$ , the minimum neighborhood problem (MNP). In this paper, we obtain $\theta _{AG_{n}}(q)$ for an independent set with size $q$ in an $n$ -dimensional alternating group graph $AG_{n}$ , a well-known interconnection network for multiprocessor systems. We first propose some combinatorial properties of $AG_{n}$ . Then, we study the MNP for an independent set of two vertices and obtain that $\theta _{AG_{n}}(2)=4n-10$ . Next, we prove that $\theta _{AG_{n}}(3)=6n-16$ . Finally, we propose that $\theta _{AG_{n}}(4)=8n-24$ .

Hybrid fault diagnosis capability analysis of hypercubes under the PMC model and MM⁎ model

System level diagnosis is an important approach for the fault diagnosis of multiprocessor systems. In system level diagnosis, diagnosability is an important measure of the diagnosis capability of interconnection networks. But as a measure, diagnosability can not reflect the diagnosis capability of multiprocessor systems to link faults which may occur in real circumstances. In this paper, we propose the definition of h-edge tolerable diagnosability to better measure the diagnosis capability of interconnection networks under hybrid fault circumstances. The h-edge tolerable diagnosability of a multiprocessor system G is the maximum number of faulty nodes that the system can guarantee to locate when the number of faulty edges does not exceed h, denoted by the(G). The PMC model and MM model are the two most widely studied diagnosis models for the system level diagnosis of multiprocessor systems. The hypercubes are the most well-known interconnection networks. In this paper, the h-edge tolerable diagnosability of n-dimensional hypercube under the PMC and MM⁎ model is determined as follows: the(Qn)=n−h, where 1≤h<n, n≥4.

The Conditional Diagnosability of Exchanged Crossed Cube

Conditional diagnosability is widely accepted as an important measure in determining the reliability of an interconnection network. The conditional diagnosability of many well-known interconnection networks has been investigated. Exchanged crossed cube(ECQ(s,t)) is a novel variant of hypercube, which retains the advantages of exchanged hypercube and crossed cube in terms of the smaller diameter, fewer links, and lower cost factor, and indicates more balanced consideration among performance and cost. In this paper, several topological properties of ECQ(s,t) are derived. On this basis, the conditional diagnosability of ECQ(s,t) under the PMC model is shown to be $4(s-1)+1$ for $t\ge s>2$ , which is almost two times larger than its classical diagnosability and also is larger than its conditional diagnosability under the MM model.

Intermittent Fault Diagnosability of Interconnection Networks

An interconnection network’s diagnosability is an important metric for measuring its self-diagnostic capability. Permanent fault and intermittent fault are two different fault models that exist in an interconnection network. In this paper, we focus on the problem pertaining to the diagnosability of interconnection networks in an intermittent fault situation. First, we study a class of interconnection networks called crisp three-cycle networks, in which the cn in -number (the number of common vertices each pair of vertices share) is no more than one. Necessary and sufficient conditions are derived for the diagnosability of crisp three-cycle networks under the PMC (Preparata, Metze, and Chien) model. A simple check can show that many well-known interconnection networks are crisp three-cycle networks. Second, we prove that an interconnection network S is a t i -fault diagnosable system without repair if and only if its minimum in-degree is greater than t i under the BGM (Barsi, Grandoni, and Masetrini) model. Finally, we extend the necessary and sufficient conditions to determine whether an interconnection network S is t i -fault diagnosable without repair under the MM (Maeng and Malek) model from the permanent fault situation to the intermittent fault situation.

The -Diagnosability for Regular Networks

The $t/k$ -diagnosis strategy can significantly enhance the system’s self-diagnosing capability at the expense of no more than $k$ fault-free processors (vertices) being mistakenly diagnosed as faulty under the PMC model. It is a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis on multiprocessor systems. It can detect up to $t$ faulty processors (vertices) which might include at most $k$ misdiagnosed processors (vertices), where $k$ is typically a small number. In the case $k\ge 1$ , to our knowledge, there is no known $t/k$ -diagnosis algorithm for general regular networks. In this paper, we first propose a general $t/k$ -diagnosis ( $k\ge 1$ ) algorithm for some $m$ -regular networks. These $m$ -regular networks satisfying some conditions could establish the $t/k$ -diagnosis algorithm, say $t/k$ - $G$ - $DIAG$ , to determine the $t/k$ -diagnosability. The complexity of this algorithm is only $O(N\log N)$ (when $N\ge 2^m$ ) or $O(Nm)$ (when $N ) where $N$ is the number of vertices in the network. Second, we present a complete proof that the network $G$ is actually $t/k$ -diagnosable. Finally, we establish the $t/k$ -diagnosability ( $1\le k\le 3$ ) of some regular networks, including an $n$ -dimensional alternating group graph , an $n$ -dimensional Split-Star Network , a $l^n$ -hypermesh and an $(n,l)$ -star graph , which are well-known interconnection networks proposed for multiprocessor systems.

Formula omitted]-restricted connectivity of locally twisted cubes

Given a graph G and a non-negative integer h, the h-restricted connectivity of G, denoted by κh(G), is defined as the minimum size of a set X of nodes in G (X⊂V(G)) such that G−X is disconnected, and the degree of each component in G−X is at least h. The h-restricted connectivity measure is a generalization of the traditional connectivity measure, and it improves the connectivity measurement accuracy. Moreover, studies have revealed that if a network possesses a restricted connectivity property, it is more reliable and demonstrates a lower node failure rate compared with other networks. The n-dimensional locally twisted cube LTQn, which is a well-known interconnection network for parallel computing, is a variant of the hypercube Qn. Most studies have examined the h-restricted connectivity of networks under the conditions of h=1 or h=2. This paper examines a generalized h-restricted connectivity measure for n-dimensional locally twisted cube and reveals that κh(LTQn)=2h(n−h) for 0≤h≤n−2.

The Extra Connectivity, Extra Conditional Diagnosability, and &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt; $t/m$&lt;/tex-math&gt; &lt;/inline-formula&gt;-Diagnosability of Arrangement Graphs

Extra connectivity is an important indicator of the robustness of a multiprocessor system in presence of failing processors. The $g$ -extra conditional diagnosability and the $t/m$ -diagnosability are two important diagnostic strategies at system-level that can significantly enhance the system's self-diagnosing capability. 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. The $t/m$ -diagnosis strategy can detect up to $t$ faulty processors which might include at most $m$ misdiagnosed processors, where $m$ is typically a small integer number. In this paper, we analyze the combinatorial properties and fault tolerant ability for an $(n,k)$ -arrangement graph, denoted by $A_{n,k}$ , a well-known interconnection network proposed for multiprocessor systems. We first establish that the $A_{n,k}$ 's one-extra connectivity is $(2k-1) (n-k)-1$ ( $k\geq 3$ , $n\geq k+2$ ), two-extra connectivity is $(3k-2)(n-k)-3$ ( $k\geq 4$ , $n\geq k+2$ ), and three-extra connectivity is $(4k-4)(n-k)-4$ ( $k\geq 4$ , $n\geq k+2$ or $k\geq 3$ , $n\geq k+3$ ), respectively. And then, we address the $g$ -extra conditional diagnosability of $A_{n,k}$ under the PMC model for $1\leq g \leq 3$ . Finally, we determine that the $(n,k)$ -arrangement graph $A_{n,k}$ is $[(2k-1)(n-k)-1]/1$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), $[(3k-2)(n-k)-3]/2$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), and $[(4k-4)(n-k)-4]/3$ -diagnosable ( $k\geq 4$ , $n\geq k+3$ ) under the PMC model, respectively.

The extra connectivity of bubble-sort star graphs

Connectivity plays an important role in measuring the fault tolerance of a multiprocessor system in the case of vertices failures. Extra connectivity is an important indicator of a network's ability for diagnosis and fault tolerance. In this paper, we analyze the fault tolerance ability of the bubble-sort star network, denoted by BSn, a well-known interconnection network proposed for multiprocessor systems, and establish the g-extra connectivity for 1≤g≤3.

The Extra, Restricted Connectivity and Conditional Diagnosability of Split-Star Networks

Connectivity is a classic measure for fault tolerance of a network in the case of vertices failures. Extra connectivity and restricted connectivity are two important indicators of the robustness of a multi-processor system in presence of failing processors. An interconnection network's diagnosability is an important measure of its self-diagnostic capability. The conditional diagnosability is widely accepted as a new measure of diagnosability by assuming that any fault-set cannot contain all neighbors of any node in a multiprocessor system. In this paper, we analyze the combinatorial properties and fault tolerance ability for the Split-Star Network, denoted by $S_n^2$ , a well-known interconnection network proposed for multiprocessor systems, establish the $g$ -extra connectivity, where $1\le g\le 3$ . We also determine the $h$ -restricted connectivity ( $h=1,2$ ), and prove that the conditional diagnosability of $S_n^2$ $(n\ge 4)$ is $6n-16$ under the comparison model, which is about three times of the $S_n^2$ 's traditional diagnosability. As a product, the strong diagnosability of $S_n^2$ is also 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.

Conditional diagnosability and strong diagnosability of Split-Star Networks under the PMC model

An interconnection network's diagnosability is an important measure of its self-diagnostic capability. The classical problems of fault diagnosis are explored widely. The conditional diagnosability is proposed by Lai et al. as a new measure of diagnosability, which can better measure the diagnosability of regular interconnection networks. The conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors. Furthermore, a multiprocessor system is strongly t-diagnos-able, if it is t-diagnosable and can achieve diagnosability t+1 except for the case where a node's neighbors are all faulty. The conditional diagnosability and strong diagnosability were proposed later to better reflect the networks' self-diagnostic capability under more realistic assumptions. In this paper, we determine the conditional diagnosability of an n-dimensional Split-Star Network (denoted as Sn2), a well-known interconnection network model for multiprocessor systems, under the PMC (Preparata, Metze, and Chien) model. We show that the conditional diagnosability of Sn2(n≥4) is 8n−23, which is about four times of its traditional diagnosability. As a byproduct, the strong diagnosability of Sn2 is also obtained.

Edge-fault-tolerant pancyclicity of arrangement graphs

The arrangement graph An,k is a well-known interconnection network. Day and Tripathi proved that An,k is pancyclic for n-k⩾2. In this paper, we improve this result, and we demonstrate that An,k is also pancyclic even if it has no more than (k(n-k)-2) faulty edges for n-k⩾2. Our result is optimal concerning the edge fault-tolerance.

Edge-fault tolerance of hypercube-like networks

This paper considers a kind of generalized measure λs(h) of fault tolerance in a hypercube-like graph Gn which contains several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes, Möbius cubes and the recursive circulant G(2n,4), and proves λs(h)(Gn)=2h(n−h) for any h with 0≤h≤n−1 by the induction on n and a new technique. This result shows that at least 2h(n−h) edges of Gn have to be removed to get a disconnected graph that contains no vertices of degree less than h. Compared with previous results, this result enhances fault-tolerant ability of the above-mentioned networks theoretically.

The Conditional Diagnosability of k-Ary n-Cubes under the Comparison Diagnosis Model

Processor fault diagnosis plays an important role in measuring the reliability of multiprocessor systems and diagnosing many well-known interconnection networks. Conditional diagnosability is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. This study investigates some topological properties of k-ary n-cubes, where k ≥ 4 and n ≥ 4, and shows that the conditional diagnosability of k-ary n-cubes under the comparison diagnosis model is 6n - 5.