Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

2021 ◽  
pp. 1-26
Author(s):  
Yihong Wang ◽  
Cheng-Kuan Lin ◽  
Shuming Zhou ◽  
Tao Tian
Keyword(s):  
Big Data ◽  
Fault Tolerance ◽  
Connected Graph ◽  
Interconnection Networks ◽  
Large Scale ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Multiprocessor Systems ◽  
Edge Disjoint ◽  
Cube Formula

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph [Formula: see text] is called strong Menger (edge) connected if, for any two distinct vertices [Formula: see text] and [Formula: see text], there are [Formula: see text] vertex (edge)-disjoint paths between them. Exchanged hypercube [Formula: see text], as a variant of hypercube [Formula: see text], remains lots of preferable fault tolerant properties of hypercube. In this paper, we show that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are strong Menger (edge) connected, respectively. Moreover, as a by-product, for dual cube [Formula: see text], one popular generalization of hypercube, [Formula: see text] is also showed to be strong Menger (edge) connected, where [Formula: see text].

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

2019 ◽  
Vol 30 (08) ◽  
pp. 1301-1315 ◽  
Author(s):  
Liqiong Xu ◽  
Shuming Zhou ◽  
Weihua Yang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Local Connectivity ◽  
Communication Links ◽  
Restricted Condition ◽  
Vertex Disjoint

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.

scholarly journals Strong Menger Connectedness of Augmented k-ary n-cubes

2020 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Arbitrary Vertex ◽  
Edge Disjoint ◽  
Vertex Set ◽  
Delta G ◽  
Disjoint Edge

Abstract A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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

2020 ◽  
Vol 30 (03) ◽  
pp. 2040001
Author(s):  
Liyang Zhai ◽  
Liqiong Xu ◽  
Weihua Yang
Keyword(s):  
Regular Graph ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Disjoint Paths ◽  
Communication Links ◽  
Restricted Condition ◽  
Local Edge ◽  
Augmented Cube ◽  
Cube Formula

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.

Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

2021 ◽  
pp. 1-12
Author(s):  
Yuxing Yang
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Text Structure ◽  
Multiprocessor Systems ◽  
Cube Formula

Let [Formula: see text] be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super [Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

Fault-tolerant strategy and workspace of the subreflector parallel adjusting mechanism

2019 ◽  
Vol 233 (18) ◽  
pp. 6656-6667
Author(s):  
Jiantao Yao ◽  
Bo Han ◽  
Yuchao Dou ◽  
Yundou Xu ◽  
Yongsheng Zhao
Keyword(s):  
Fault Tolerance ◽  
Large Scale ◽  
Fault Tolerant ◽  
Degree Of Freedom ◽  
Proper Function ◽  
Maintenance Cost ◽  
Mechanical Equipment ◽  
Identification Method ◽  
Boundary Identification ◽  
Workspace Boundary

Parallel mechanism has been widely used in large-scale and heavy-duty attitude adjustment equipment. In order to improve the work reliability of the subreflector parallel adjusting mechanism, a fault-tolerant strategy based on the redundant degree-of-freedom and a workspace boundary identification method are proposed in this paper, which can realize the proper function of the subreflector parallel adjusting mechanism when it has a driven fault. The configuration and parameters of the parallel adjusting mechanism are introduced firstly, then the degrees-of-freedom of the parallel adjusting mechanism is calculated when it has a driven fault, and the principle of the fault-tolerant strategy based on the redundant degree-of-freedom is deduced in detail. Next, the method to solve the workspace boundary identification problem for the parallel adjusting mechanism in fault tolerance conditions is proposed, the maximum and minimum workspaces of the parallel adjusting mechanism at fault tolerance conditions in different frequency bands are analyzed. The results showed that the workspace calculated by the fault-tolerant strategy in the fault condition can completely meet the needs of the subreflector, where this method can also be applied to other parallel mechanisms. Lastly, an experiment is conducted to verify the correctness and effectiveness of the fault-tolerant strategy, in which the results showed that the fault-tolerant strategy can effectively improve the work reliability of the parallel adjusting mechanism. The fault-tolerant strategy and workspace boundary identification method can make the subreflector parallel adjusting mechanism work normally when it has a driven fault, which can significantly improve the work reliability and work efficiency and the maintenance cost can also be reduced. The fault-tolerant strategy and workspace boundary identification method can also be well applied to the research and development for this kind of parallel mechanical equipment.

K-PROCESSOR RELIABILITY OF LARGE-SCALE RING-BASED HIERARCHICAL INTERCONNECTIONS

2002 ◽  
Vol 09 (01) ◽  
pp. 61-77
Author(s):  
M. AL-ROUSAN ◽  
O. AL-JARRAH ◽  
M. MOWAFI
Keyword(s):  
Interconnection Networks ◽  
Large Scale ◽  
Network Reliability ◽  
Multiprocessor Systems ◽  
Hierarchical Systems ◽  
Hierarchical Networks ◽  
Coherent Interface ◽  
Large Acceptance ◽  
Ring Architecture ◽  
Scalable Coherent Interface

Recently, connecting thousands of processors via interconnection networks based on multiple (hierarchical) rings has an increased interest. This is due to the large acceptance and success of the Scalable Coherent Interface (SCI) technology. The inherently weak behavior of ring architecture has led interconnection designers to consider various choices to improve the overall network reliability. An interesting choice is to use braided rings instead of the single (basic) rings in the hierarchy. In this paper, we present new formulas for computing K-processor reliability of SCI ring-based hierarchical networks in the context of large-scale multiprocessor systems. The derived formulas are general and applicable to any given systems size consisting of an arbitrary number of levels. The reliability of hierarchical systems based on the basic and braided rings is evaluated and analyzed using the derived formulas. The results show that hierarchical systems based on braided rings significantly improve the reliability of hierarchies constructed of basic rings. The results are general and not limited to systems of SCI rings; the analysis is valid for any type of rings architecture such as token and slotted rings.

Designing a New Class of Fault Tolerant Multistage Interconnection Networks

2005 ◽  
Vol 06 (04) ◽  
pp. 361-382 ◽  
Author(s):  
K. V. Arya ◽  
R. K. Ghosh
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Multistage Interconnection Networks ◽  
Multistage Interconnection Network ◽  
New Class

This paper proposes a technique to modify a Multistage Interconnection Network (MIN) to augment it with fault tolerant capabilities. The augmented MIN is referred to as Enhanced MIN (E-MIN). The technique employed for construction of E-MIN is compared with the two known physical fault tolerance techniques, namely, extra staging and chaining. EMINs are found to be more generic than extra staged networks and less expensive than chained networks. The EMIN realizes all the permutations realizable by the original MIN. The routing strategies under faulty and fault free conditions are shown to be very simple in the case of E-MINs.

FAULT-TOLERANT COMMUNICATIONS ON HYPERCUBE-CLUSTERS

2000 ◽  
Vol 01 (04) ◽  
pp. 315-329 ◽  
Author(s):  
PETER KOK KEONG LOH ◽  
WEN JING HSU
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Routing Algorithm ◽  
Cost Effective ◽  
Hardware Support ◽  
Tolerance Level ◽  
Component Faults ◽  
Message Overhead ◽  
Standard Components

Hierarchical interconnection networks with n-dimensional hypercube clusters can strike a balance between wide application suitability, size scalability as well as reliability. Cluster communications support for such networks must therefore be reliable and efficient without incurring large overheads. This paper proposes a reliable and cost-effective intra-cluster communications strategy for such a class of interconnection networks. The routing algorithm can tolerate up to (n - 1) component faults in the cluster and generates routes that are cycle-free and livelock-free. The message is guaranteed to be optimally (respectively, sub-optimally) delivered within a maximum of n (respectively, 2n - 1) hops. The message overhead incurred is one of the lowest reported for the specified fault tolerance level – with only a single n-bit routing vector accompanying the message to be communicated. Finally, routing hardware support may be simply achieved with standard components, facilitating integration with the host network.

Structure Fault Tolerance of Recursive Interconnection Networks

2020 ◽  
Author(s):  
Eminjan Sabir ◽  
Jixiang Meng
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Theoretical Problem ◽  
Connected Subgraph ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Edge Disjoint ◽  
Balanced Hypercube

Abstract Motivated by effects caused by structure link faults in networks, we study the following graph theoretical problem. Let $T$ be a connected subgraph of a graph $G$ except for $K_{1}$. The $T$-structure edge-connectivity $\lambda (G;T)$ (resp. $T$-substructure edge-connectivity $\lambda ^s(G;T)$) of $G$ is the minimum cardinality of a set of edge-disjoint subgraphs $\mathcal{F}=\{T_{1},T_{2},\ldots ,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},\ldots ,T_{m}^{^{\prime}}\}$) such that $T_{i}$ is isomorphic to $T$ (resp. $T_{i}^{^{\prime}}$ is a connected subgraph of $T$) for every $1 \le i \le m$, and $E(\mathcal{F})$’s removal leaves the remaining graph disconnected. In this paper, we determine both $\lambda (G;T)$ and $\lambda ^{s}(G;T)$ for $(1)$ the hypercube $Q_{n}$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},P_{4},Q_{1},Q_{2},Q_{3}\}$; $(2)$ the $k$-ary $n$-cube $Q^{k}_{n}$$(k\ge 3)$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},Q^{3}_{1},Q^{4}_{1}\}$; $(3)$ the balanced hypercube $BH_{n}$ and $T\in \{K_{1,1},K_{1,2},BH_{1}\}$. We also extend some known results.

scholarly journals Dynamic Fault Tolerant Topology Control for Wireless Sensor Network Based on Node Cascading Failure

2018 ◽  
Vol 14 (05) ◽  
pp. 118 ◽  
Author(s):  
Yang Xiao
Keyword(s):  
Fault Tolerance ◽  
Degree Distribution ◽  
Large Scale ◽  
Fault Tolerant ◽  
Wireless Sensor ◽  
Cascading Failure ◽  
Network Node ◽  
Simulation Test ◽  
Node Failure ◽  
Scale Free

To address the node cascading failure (CF) of the wireless sensor networks (WSNs), considering such factors as node load and maximum capacity in scale-free topology, this paper establishes the WSN dynamic fault tolerant topology model based on node cascading failure, analyses the relationships between node load, topology and dynamic fault tolerance, and demonstrates the proposed model through simulation test. It studies the effects of topology parameter and load in case of random node failure in the network node cascading failure, and utilizes the theoretical derivation method to derive the structural feature of scale-free topology and the capacity limit for the WSNs large-scale cascading failure, effectively enhancing the cascading fault tolerance of traditional WSNs. The simulation test results show that, with the degree distribution parameter <em>C</em> increasing, the minimum network node degree will increase accordingly, and in highly intensive topology, the dynamic fault tolerance will be better; with the parameter<em> λ </em>increasing, the maximum degree of the network node will gradually decrease, and the degree distribution of topology structure tends to be uniform, so that the large-scale cascading failure caused by node failure will have less influence on the WSN, and further improve the dynamic fault tolerance performance of the system.

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