A network can contain numerous spanning trees. If two spanning trees T i , T j do not share any common edges, T i and T j are said to be pairwisely edge-disjoint. For spanning trees T 1 , T 2 ,…, T m , if every two of them are pairwisely edge-disjoint, they are called completely edge-independent spanning trees (CEISTs for short). CEISTs can facilitate many network functionalities, and constructing CEISTs as maximally allowed as possible in a given network is a worthy undertaking. In this paper, we establish the maximal number of CEISTs in the locally twisted cube network, and propose an algorithm to construct ⌊ n 2 ⌋ CEISTs in LTQ n , the n -dimensional locally twisted cube. The proposed algorithm has been actually implemented, and we present the outputs. Network broadcasting in the LTQ n was simulated using ⌊ n 2 ⌋ CEISTs, and the performance compared with broadcasting using a single tree.

It is widely known that parallel and distributed systems are crucial technologies and platforms necessary to support supercomputing and cloud computing. The network architecture forms the foundational support for the stable operation of these systems, directly influencing their reliability, scalability, and robustness. As the network scale expands, the probability of processor/server and communication link failures increases. Therefore, it is imminent to consider how to build up the fault tolerance and reliability of the network. The circulant-based recursive networks (CRNs) are a novel type of network with several desirable properties such as regularity, recursiveness, vertex (edge) transitivity and so on. CRNs contain not only interconnection networks hypercubes and$k$-ary$n$-cubes, but also data center network BCube, as well as some future networks. Connectivity and diagnosability of networks have garnered significant attention, as they suffice for analyzing and measuring networks' fault tolerance. This article focuses primarily on conditional connectivity and diagnosability under the good neighbor fault pattern. In this work, we explore the conditional connectivity and diagnosability (built on$g$-good neighbor fault pattern) of the$f$-dimensional$r$-order CRN under the PMC model and MM* model, respectively. These values are nearly$g$times greater than the traditional connectivity and diagnosability of CRNs, respectively, implying that they can further improve fault tolerance of CRNs. Furthermore, it is worth noting that the results can be effectively utilized in BCube and other future networks given that they are both subclasses of CRNs.

With the increasing network scale, hardware faults are inevitable. Therefore, research on network reliability under the condition of hardware failure is an important subject. In this article, we investigate$h$-extra connectivity,$h$-extra diagnosability under the PMC and MM$^{*}$models, and$t/m$-diagnosability under the PMC model of recursive networks based on complete graphs (RNCGs) that include not only interconnection network Dragonfly but also data center networks—DCell, generalized DCell, and other unknown networks. In addition, we propose the fault-tolerant routing (F-TR) algorithm, FTPath, which constructs a F-TR in the largest component when the number of faulty nodes is less than the$h$-extra connectivity. Moreover, we evaluate the performance of RNCGs. First, we compare the fault-tolerant performance of RNCGs through experiments. The results show that the average path length constructed by the FTPath algorithm is close to that of the breadth-first search algorithm, and shorter than that of the depth-first search algorithm. Second, we evaluate the performance of this network for different parameters:$h$-extra connectivity, diagnosability under different strategies. The results show that the network has good fault tolerance and fault diagnosis capabilities.

Abstract According to Menger’s theorem, connectivity and edge connectivity are closely related to node-disjoint paths and edge-disjoint paths, respectively. Node- and edge-disjoint paths can keep the effective transmission of information and confidentiality. Therefore, node-disjoint paths and edge-disjoint paths are two important parameters to measure the reliability of a network. For a faulty node (resp. edge) set $S\subset V$ (resp. $S\subset E$), a connected graph $G=(V,E)$ is $S$-strongly Menger-node-connected (resp. Menger-edge-connected) if any two distinct nodes $x$ and $y$ in $G-S$ are connected by $\min \{\deg _{G-S}(x),\deg _{G-S}(y)\}$ internally node-disjoint (resp. edge-disjoint) paths in $G-S$, where $\deg _{G-S}(x)$ and $\deg _{G-S}(y)$ are the degrees of $x$ and $y$ in $G-S$, respectively. And most of the previous studies are based on networks that are triangle-free. In this paper, we consider the strongly Menger (edge) connectedness of a class of $r$-dimensional recursive networks RNCG $G_{r}$ with triangles. Moreover, we show that $G_{r}$ is $(rl-l-1)$-strongly Menger-node-connected. And then we show that $G_{r}$ is $[k+(r-1)l-2]$-strongly Menger-edge-connected of order 1 and $[2k+2(r-1)l-6]$-strongly Menger-edge-connected of order 2. Since the class of $r$-dimensional recursive networks RNCG $G_{r}$ includes not only data center networks DCell and generalized DCell but also interconnection network dragonfly, etc., all the results are appropriate for these networks.

Networks of large scales are an essential component in supercomputing systems as well as in data centers. As the network scale increases, the probability of processor/server failures also inevitably increases. It is therefore a worthwhile undertaking to make efforts reducing, as much as possible, the effect of faulty processors/servers to the entire network. This paper introduces a new class of network architectures, called circulant-based recursive networks (CRNs), and investigates CRN’s diameter, connectivity, and in particular, the fault diagnosability under the two diagnostic models−the PMC and the comparison diagnostic models. CRNs are a generalization of some well-known interconnection networks−hypercube, k-ary n-cube network and the data center network BCube, as well as some other less-known networks. In addition to obtaining its diagnosability properties, the paper also presents a one-to-one (unicast) path construction algorithm named SPath. Based on SPath, we further propose an algorithm FTPath for CRNs finding a fault-tolerant path between any two vertices, provided that the number of faulty vertices is no more than its connectivity minus one. Three parameters−average distance, message density, and cost−are used to assess CRNs’ performance. Experimental comparisons are conducted, and the results indicate that the average path length obtained by the algorithm SPath (resp., FTPath) is shorter than that of the Depth-First Search algorithm (DFS) and is on a par with the Breath-First Search algorithm (BFS).

One of the important issues in evaluating an interconnection network is to study the fault-tolerant Hamiltonian cycle and Hamiltonian path embedding problems. The$k$-ary$n$-cube (denoted by$Q^{k}_{n}$) networks are used as interconnection networks for many parallel and distributed computing systems. In this article, we investigate the Hamiltonian cycle and path embeddings in the bipartite$k$-ary$n$-cube$Q^{k}_{n}$based on$K_{1,1}$-structure faults. We show that there exists a Hamiltonian cycle in$Q^{k}_{n}-\mathcal {F}$if$|\mathcal {F}|\leq 2n-2$and there exists a Hamiltonian path between any two vertices from different partite sets in$Q^{k}_{n}-\mathcal {F}$if$|\mathcal {F}|\leq 2n-3$for$n\geq 2$and even$k\geq 4$, where$\mathcal {F}$is a set of vertex-disjoint subgraphs isomorphic to$K_{1,1}$in$Q_{n}^{k}$. In some sense, the results mean that when a subset$S$of at most$4n-4$(resp.$4n-6$) processors is deleted from a bipartite$Q^{k}_{n}$, there exists a Hamiltonian cycle (resp. a Hamiltonian path between any two healthy processors from different partite sets) in the remaining network. Our results, in some sense, compensate the results in Lv et al. [J. Parallel Distrib. Comput., 120, 148–158, 2018] and [Comput. J., 60, 159–179, 2017], where authors studied the$K_{1,3}$-substructure fault-tolerant Hamiltonian cycle and path embedding problems in nonbipartite$k$-ary$n$-cubes. In comparison, the bipartite$k$-ary$n$-cube$Q^{k}_{n}$can keep the same$K_{1,1}$-structure fault-tolerant Hamiltonian capabilities as the nonbipartite one.

The efficiency of cloud computing is significantly impacted by the capabilities of data center networks (DCNs). However, with the increasing demands of applications, the number of servers in DCNs is growing exponentially. In general, networks become more vulnerable as they expand in size. Consequently, it is essential to assess the network's reliability. Subsystem reliability, a crucial parameter of network reliability, is the likelihood that a faultless subsystem with a specific size would function normally. Even though several networks have the same order, their subsystem reliability may differ. In this article, we compare two distinct subsystems of$m$-port$k$-dimensional DCNs BCube that have the same number of vertices and ascertain the critical time so that the reliability bound is robust. In addition, we discover that, when comparing two subsystems of BCube that have the same number of vertices, the smaller the dimension$k$, the greater the subsystem reliability of BCube will be for$m\geq 2$and$k\geq 1$. This offers a theoretical foundation that allows us to choose a more reliable system in BCube that have the same number of vertices. Finally, we use numerical experiments to verify our findings.

Abstract The data center network HSDC is a superior candidate for building large-scale data centers, and strikes a good balance among diameter, bisection width, incremental scalability and other important characteristics in contrast to the state-of-the-art data center network architectures. The Hamiltonian property is an important indicator to measure the reliability of a network. In this paper, we study the Hamiltonian properties of HSDC’s logic graph $H_n$. Firstly, we prove that $H_n$ is Hamiltonian-connected for $n\geq 3$. Secondly, we propose an $O(NlogN)$ algorithm for finding a Hamiltonian path between any two distinct nodes in $H_n$, where $N$ is the number of nodes in $H_n$. Furthermore, we consider the Hamiltonian properties of $H_n$ with faulty elements, and prove that $H_n$ is $(n-3)$-fault-tolerant Hamiltonian-connected and $(n-2)$-fault-tolerant Hamiltonian for $n\geq 3$.

Abstract As a kind of conditional connectivity, component connectivity is an improvement of traditional connectivity, which is conducive to enhance the reliability of the network. To be specific, the $r$-component connectivity of a network $G$, written as $c\kappa _{r}(G)$, is defined as the minimum number of all node cuts whose removal causes the remaining network to have at least $r$ components. Component diagnosability, as another measure of network reliability, is usually related to the number of components in the remaining network. The $r$-component diagnosability, written as $ct_{r}(G)$, is defined as the maximum number of faulty sets such that at least $r$ components in the surviving network and all faulty nodes can be diagnosed. This paper mainly explores the relationship between component connectivity and component diagnosability of some regular networks. Once knowing the component connectivity of such a network, with the help of this relationship, we can easily obtain the component diagnosability of the network. Furthermore, we apply this relationship to some famous regular networks to obtain their component diagnosabilities under the PMC model.

The fault tolerance of a network can be measured by many parameters. Connectivity is a classic measurement parameter for evaluating the fault tolerance of a network. g-extra connectivity and g-restricted connectivity are generalizations of connectivity, which can better reflect the fault tolerance of a network. Specifically, the g-extra connectivity $\kappa_{g}(G)$ of a graph G is the minimum number of nodes whose removal will disconnect G, and each remaining component has no less than $g+1$ nodes. Furthermore, the g-restricted connectivity $\kappa^{g}(G)$ of G is the minimum number of nodes whose deletion results in a graph being disconnected and the minimum degree of each remaining component is at least g. In general, g-restricted connectivity is not equal to g-extra connectivity of a network. Therefore, many scholars often discuss g-restricted connectivity and g-extra connectivity with regard to different networks separately. In this paper, we show that g-restricted connectivity is equal to g-extra connectivity under some conditions. Then, the relationship we derived can be applied to some known networks such as the data center networks DCell and BCDC, multiprocessor network $(n,k)$-star. In addition, we construct a new network $H(G_{0},G_{1},G_{2};\mathbb{M})$ and prove that our result can be applied to it. In detail, we prove $\kappa^{g}(H(G_{0},G_{1},G_{2};\mathbb{M}))=\kappa_{g}(H(G_{0},G_{1},G_{2};\mathbb{M}))=n+g+1$ for any integers $n\geq 3$ and $ g\displaystyle \leq\lfloor\frac{n-2}{2}\rfloor$.

Abstract In the past few years, much importance and attention have been attached to completely independent spanning trees (CISTs). Many results, such as edge-disjoint Hamilton cycles, traceability, number of spanning trees, structural properties, topological indices, etc., have been obtained on line graphs, and researchers have applied the line graphs of some interconnection networks such as generalized hypercubes, augmented cubes, crossed cubes, etc., into data center networks, such as SWCube, AQLCube, BCDC, etc. At the meanwhile, few results of CISTs are reported on the line graphs. In this paper, we establish the relation of edge-disjoint spanning trees in an interconnection network $G$’ with its line graph $G$ by proposing a general algorithm for the first time. By this method, more CISTs can be obtained comparing with results in the literature. Then, the decrease of diameter is discussed and simulation experiments are shown on the line graphs of hypercubes.

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 $n\geq 4$.

The past decade has seen growing importance being attached to theCompletely Independent Spanning Trees(CISTs). The CISTs can facilitate many network functionalities, and the existence and construction schemes of CISTs in various networks can be an indicator of the network's robustness. In this paper, we establish the number of CISTs that can be constructed in theline graphof the complete graph$K_n$(denoted$L(K_n)$, for$n\geq 4$), and present an algorithm to construct the optimal (i.e., maximal) number of CISTs in$L(K_n)$. The$L(K_n)$is a special class of SWCube [13], an architectural model proposed for data center networks. Our construction algorithm is also implemented to verify its validity.

In the past few years, edge-disjoint spanning trees (EDSTs) have attracted extensive attention due to their applications in reliable communication, fault-tolerant broadcasting, secure message distribution, etc. As one architecture of many interconnection networks, hypercubes (denoted as Qn) play an important role in parallel computing systems, as well as their line graphs (denoted as L(Qn)), but few results of EDSTs in L(Qn) are reported. In this paper, we establish the relation between EDSTs in Qnand EDSTs in L(Qn), then we propose an algorithm to obtain the EDSTs in L(Qn) and present the corresponding simulation experiment to verify its validity.

Abstract Connectivity is a classic measure for reliability of a multiprocessor system in the case of processor failures. Extra connectivity and component connectivity are two important indicators of the reliability of a multiprocessor system in presence of failing processors. The $h$-extra connectivity $\kappa _{h}(G)$ of a graph $G$ is the minimum number of nodes whose removal will disconnect $G$, and every remaining component has at least $h+1$ nodes. Moreover, the $h$-component connectivity $c\kappa _{h}(G)$ of $G$ is the minimum number of nodes whose deletion results in a graph with at least $h$ components. However, the extra connectivity and component connectivity of many well-known networks have been independently investigated. In this paper, we determine the relationship between extra connectivity and component connectivity of general networks. As applications, the extra connectivity and component connectivity are explored for some well-known networks, including complete cubic networks, hierarchical cubic networks, generalized exchanged hypercubes, dual-cube-like networks, Cayley graphs generated by transposition trees and hierarchical hypercubes as well.

Abstract As the number of servers in a data center network (DCN) increases, the probability of server failures is significantly increased. Traditional connectivity is an important metric to measure the reliability of DCN. However, the traditional connectivity of a DCN based on the condition of arbitrary faulty servers is generally lower. Therefore, it is important to increase the connectivity of a DCN by adding some limited conditions for the faulty server set. As a result, $g$-restricted connectivity and $h$-extra connectivity, which are two crucial subjects for a DCN’s ability to tolerate faulty servers, were proposed in the literature. In this paper, we study the $g$-restricted connectivity and $h$-extra connectivity of a new server-centric DCN, called BCDC, based on crossed cube with excellent performance. We prove that the $g$-restricted connectivity of BCDC is 4 for $n=3$ and $2n+g(n-2)-2$ for $n\geq 4$, where $0\leq g\leq n-3$, and the $h$-extra connectivity of BCDC is 4 for $n=3$ and $2n+h(n-2)-2$ for $n\geq 4$, where $0\leq h\leq n-3$.

A set of spanning trees T1; T2;. .. ; Tk in a network G are Completely Independent Spanning Trees (CISTs) if for any two nodes u and v in V (G), the paths between u and v in any two trees have no common edges and no common internal nodes. CISTs have important applications in data center networks, such as fault-tolerant multi-node broadcasting, fault-tolerant one-to-all broadcasting, reliable broadcasting, secure message distribution, and so on. The augmented cube AQn is a prominent variant of the well-known hypercube Qn, and having the important property of scalability, and both Qn and AQn have been proposed as the underlying structure for a data center network. The data center network based on AQn is denoted by AQDNn, and the logic graph of AQDNn is denoted by L-AQDNn. In this article, we study how to construct n - 1 CISTs in L-AQDNn. The constructed n - 1 CISTs are optimal in the sense that n - 1 is the maximally allowed CISTs in L-AQDNn. The correctness of our construction algorithm is proved. It is the first time a direct relationship is established between the dimension of a hypercube-family network and the number of CISTs it can host.

For a network, edge/node-independent spanning trees (ISTs) can not only tolerate faulty edges/nodes, but also be used to distribute secure messages. As important node-symmetric variants of the hypercubes, the augmented cubes have received much attention from researchers. The n-dimensional augmented cube AQn is both (2n ‒ 1)-edge-connected and (2n ‒ 1)-nodeconnected (n ≢ 3), thus the well-known edge conjecture and node conjecture of ISTs are both interesting questions in AQn. So far, the edge conjecture on augmented cubes was proved to be true. However, the node conjecture on AQn is still open. In this paper, we further study the construction principle of the node-ISTs by using the double neighbors of every node in the higher dimension. We prove the existence of 2k − 1 node-ISTs rooted at node 0 in A Q n ( 00...0 ︸ n−k )(n≥k≥4) by proposing an ingenious way of construction and propose a corresponding O(NlogN) time algorithm, where N = 2k is the number of nodes in A Q n ( 00...0 ︸ n−k ) .

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-ary 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: LDTEGH and LDWBEGH 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.

Fault-tolerance is an important parameter to measure the performance of a network. However, most works only consider the fault of single vertex and ignore the structure-fault of a network. The generalized hypercube G(mr,mr−1,…,m1) is one key interconnection network with excellent topological properties. In this paper, we study the H-structure fault-tolerance of G(mr,mr−1,…,m1) network by studying its H-structure connectivity and H-substructure connectivity for H∈{K1,M,C3,C4,K4}⁠. Since the generalized hypercube network can be used to construct some data center networks, such as BCube, HyperX, and FBFLY, the results in this paper can be applied not only to interconnection networks but also to data center networks.

Summary With the increasing number of users and a huge scale of data, the service providers of Online Social Networks (OSNs) are facing the problem of how to place users' data to multiple servers. Key‐value stores solve the problem based on consistent hashing, and have become a defacto standard. However, random placement manner of hashing cannot preserve social locality, which leads to high intra‐data center traffic and unpredictable response time. Many existing works solve the problem by using graph partitioning algorithms. These works have two drawbacks: First, the social graph is constructed with ordinary pairwise graph that cannot fully reflect multi‐participant interactions often occurring in OSNs. Second, the underlying network topologies of data center have never been considered. This paper investigates the problem of traffic minimization for OSNs data storage. Motivated by maximally preserving both social locality and distance locality, we formulate the problem as two sub‐problems — hypergraph partitioning and partition‐to‐server mapping, and propose a two‐phase data placement (TDP) scheme. Specifically we present two algorithms to solve partition‐to‐server mapping over two widely used network topologies (i.e.,tree and BCube). Evaluations with a large scale Facebook trace show that TDP significantly reduces intra‐data center traffic as well as load balancing across servers. Copyright © 2016 John Wiley & Sons, Ltd.

Reliable broadcasting for a network can be obtained by using completely independent spanning trees(CISTs). Locally twisted cubes are popular networks which have been studied widely in the literature. In this paper, we study the problem of using CISTs to establish reliable broadcasting in locally twisted cubes. We first propose an algorithm, named LTQCIST, to construct two CISTs in locally twisted cubes, then exemplify the construction procedures to construct CISTs. Finally, we prove the correctness of Algorithm LTQCIST and simulate CISTs with JUNG.

The complete binary tree is an important network structure for parallel and distributed computing, which has many nice properties and is often used to be embedded into other interconnection architectures. The locally twisted cube LTQn is an important variant of the hypercube Qn. It has many better properties than Qn with the same number of edges and vertices. In this paper, we prove that the complete binary tree CBTn can be embedded with dilation 2 and congestion 1 into LTQn. Furthermore, it is proven that if there exists an arbitrary faulty node in LTQn, then both the dilation and congestion values will become 2 after reconfiguring CBTn, while if there are two arbitrary faulty nodes in LTQn, then both the dilation and congestion values will become 3 after reconfiguration.

In recent years, there are many new findings on independent spanning trees (ISTs for short) in hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, which all belong to a more general network category called bijective connection networks (BC networks). However, little progress has been made for ISTs in general BC networks. In this paper, we first propose the definitions of conditional BC networks and V -dimensional-permutation. We then give a linear parallel algorithm of ISTs rooted at an arbitrary vertex in conditional BC networks, which include hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, based on the ascending circular dimensional-permutation, where the ISTs are all isomorphic to the binomial-like tree. In addition, we show that there exists an efficient algorithm to construct a spanning tree rooted at an arbitrary vertex in any BC network Xn, and all V-dimensional-permutations can be used to construct spanning trees isomorphic to the n-level binomial tree and rooted at an arbitrary vertex in Xn.

Independent spanning trees (ISTs) on networks have applications in networks such as reliable communication protocols, the multi-node broadcasting, one-to-all broadcasting, reliable broadcasting and secure message distribution. However, there is a problem on ISTs on graphs: If a graph G is n-connected (n ≥ 1), then there are n ISTs rooted at an arbitrary vertex on G. This problem has remained open for n ≥ 5. In this paper, we consider the construction of ISTs on Möbius cubes—a class of hypercube variants. An O(N log N) recursive algorithm is proposed to construct n ISTs rooted at an arbitrary vertex on the n-dimensional Möbius cube Mn, where N = 2n is the number of vertices in Mn. Furthermore, we prove that each IST obtained by our algorithm is isomorphic to an n-level binomial-like tree with the height n + 1 for n ≥ 2.