Fault-Tolerant Secure Routing of B H n -Based Data Center Networks

As the core infrastructure of cloud computing, a large scale of the data center networks (DCNs), which consist of millions of servers with high capacity, suffer from node failure such that the reliability is deteriorated. Malicious group could inevitably compromise the quality and reliability of data; thus, how to ensure the security routing of data is an urgent practical problem. As models for large-scale DCNs, it is worth mentioning the balanced hypercube, which is well-known for its strong connectivity, regularity, and a smaller diameter. Each of which makes a balanced hypercube a trustworthy model to deal with data traffic and provides a certain degree of fault-tolerance as well. In this paper, we use the balanced hypercube as a model for the data center networks and design a reliable safety level by referring to different safety levels of related subgraph. This subgraph contains the source and destination nodes, and the shortest feasible paths are located so that the reliable transmission is achieved. Then, we get that the length of fault-tolerant safety routing of data center networks based on balanced hypercube is always no greater than the Hamming distance plus two. Experiment shows that our fault-tolerant security routing scheme is more effective in the same reliable network environment of DCNs.

Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubes

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

Structure Fault Tolerance of Recursive Interconnection Networks

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 $\mathcal{F}=\{T_{1},T_{2},...,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},...,T_{m}^{^{\prime}}\}$) 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 results in Lin, C.-K., Zhang, L., Fan, J. and Wang, D. (2016, Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107) and Lv, Y., Fan, J., Hsu, D.F. and Lin, C.-K. (2018, Structure connectivity and substructure connectivity of $k$-ary $n$-cubes. Inf. Sci., 433, 115–124).

Structure Fault Tolerance of Recursive Interconnection Networks

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.

Fault-Free Hamiltonian Cycles in Balanced Hypercubes with Conditional Edge Faults

The balanced hypercube, [Formula: see text], is a variant of hypercube [Formula: see text]. Zhou et al. [Inform. Sci. 300 (2015) 20–27] proposed an interesting problem that whether there is a fault-free Hamiltonian cycle in [Formula: see text] with each vertex incident to at least two fault-free edges. In this paper, we consider this problem and show that each fault-free edge lies on a fault-free Hamiltonian cycle in [Formula: see text] after no more than [Formula: see text] faulty edges occur if each vertex is incident with at least two fault-free edges for all [Formula: see text]. Our result is optimal with respect to the maximum number of tolerated edge faults.

How to Systematically Embed Cycles in Balanced Hypercubes

The balanced hypercube is a variant of the hypercube structure and has desirable properties like connectivity, regularity, and symmetry. The cycle is a popular interconnection topology and has been widely used in distributed-memory parallel computers. Moreover, parallel algorithms of cycles have been extensively developed and used. The problem of how to embed cycles into a host graph has attracted a great attention in recent years. However, there is no systematic method proposed to generate the desired cycles in balanced hypercubes. In this paper, the authors develop systematic linear time algorithm to construct cycles and Hamiltonian cycles for the balanced hypercube.