Variant Of Hypercube Research Articles

One-to-one node disjoint paths on divide-and-swap cubes

In massive parallel and distributed systems, high fault-tolerance and reliability are extremely important for interconnection networks. Node-disjoint paths play a key role in improving the performance, reliability and scalability of networks.The hypercube, as well as many hypercube-like variants, are among the most popular and attractive network architectures. The divide-and-swap cube DS C n is a newly proposed hypercube variant which has many desirable properties. In this work, we show that there exist d + 1 node-disjoint paths between any two distinct nodes in divide-and-swap cubes. Since the connectivity of the network is d + 1, d + 1 is the maximum number of paths that can be constructed between any two distinct nodes.

Routing and wavelength assignment for folded hypercube in linear array WDM optical networks

The folded hypercube is one of the hypercube variants and is of great significance in the study of interconnection networks. In a folded hypercube, information can be broadcast using efficient distributed algorithms. In the context of parallel computing, folded hypercube has been studied as a possible network topology as an alternative to the hypercube. The routing and wavelength assignment (RWA) problem is significant, since it improves the performance of wavelength-routed all-optical networks constructed using wavelength division multiplexing approach. Given the physical network topology, the aim of the RWA problem is to establish routes for the connection requests and assign the fewest possible wavelengths in accordance with the wavelength continuity and distinct wavelength constraints. This paper discusses the RWA problem in a linear array for the folded hypercube communication pattern by using the congestion technique.

Chromatic numbers of certain hypercube variants

The total (vertex/edge) chromatic number of a graph G, χ ″ ( G ) ( χ ( G ) / χ ′ ( G ) ) is defined to be the minimum number of colours needed to colour the vertices and edges, together termed as elements (vertices/edges) of a graph in such a way that no two adjacent/incident elements (vertices/edges) are given the same colour. In this article, we determine the vertex, edge and total chromatic number of some hypercube variants such as folded hypercube, enhanced hypercube, augmented hypercube and exchanged hypercube.

A Note of Reliability Analysis of SM-λ in Folded-Crossed Hypercube with Conditional Faults

The fault tolerance of an interconnection network of parallel and distributed systems can be evaluated by various topological parameters of its underlying graph [Formula: see text], with strong Menger edge connectivity being a vital parameter in this regard. A connected graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text]) if it connects any pair of vertices [Formula: see text] and [Formula: see text] with [Formula: see text] number of edge-disjoint paths. Under the uniform distribution of faults in a large interconnection network, it is improbable that each faulty edge incident to a vertex will occur simultaneously. Thus, [Formula: see text]-strongly Menger edge connected of order [Formula: see text] was introduced in 2018 by He et al. Here, [Formula: see text] is called as [Formula: see text]-strongly Menger edge connected of order [Formula: see text], if [Formula: see text] remains SM-[Formula: see text], where [Formula: see text] is an arbitrary edge set in a graph [Formula: see text] with [Formula: see text] and the minimum degree of the remaining graph [Formula: see text]. The largest [Formula: see text] keeping the property of [Formula: see text] being SM-[Formula: see text] is denoted as [Formula: see text]. Among variants of hypercube, the [Formula: see text]-dimensional folded-crossed hypercube [Formula: see text] attracts attention in recent years. In this paper, we focus on calculating the exact value of the maximum conditional edge-fault-tolerant number of order [Formula: see text] of [Formula: see text], [Formula: see text] for two integers [Formula: see text] and [Formula: see text].

The Generalized 3-Connectivity of Exchanged Crossed Cube

Let [Formula: see text] be a connected graph, and [Formula: see text] with [Formula: see text]. [Formula: see text] refers to the maximum number [Formula: see text] of edge-disjoint trees [Formula: see text] in [Formula: see text] such that [Formula: see text] for distinct [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text], denoted by [Formula: see text], is defined as the minimum value of [Formula: see text] over all [Formula: see text] with [Formula: see text]. In fact, [Formula: see text] is exactly the traditional connectivity of [Formula: see text]. Exchanged crossed cube [Formula: see text], a variation of hypercube, has better properties than other variations of hypercube. In this work we obtain that [Formula: see text] with [Formula: see text].

Irregular chromatic number for hypercube graph and its variants

The attractive properties of the hypercube graph such as its diameter, good connectivity, and symmetry have made it a popular topology for the design of multi-computer interconnection networks. Efforts to improve some of these properties have led to the evolution of hypercube variants. Let c be the proper coloring of graph G, where the neighboring vertices will get individual colors. Coloring c is irregular if distinct vertices have distinct color codes and the least number of colors that ought to receive an irregular coloring is the irregular chromatic number, χir (G). In this paper, we discuss the irregular coloring and find the irregular chromatic number for the hypercube graph Qn and some of its variants using binomial coefficients for the Locally twisted cube graph LTQn, Crossed cube graph CQn and two types of Fractal cubic network graph FCNG1 (k) and FCNG2 (k).

Connectivity and super connectivity of folded hypercube-like networks

We define a new class of graphs by generalizing the hypercube-like networks to folded hypercube-like networks FHn. We also investigate the reliability of this class of graphs in terms of the (edge) connectivity and super (edge) connectivity. Since this class contains two well-known hypercube variants, namely the folded hypercube and the folded crossed cube, this paper extends some previously known results.

Maximal matching cover pebbling number for variants of hypercube

An edge pebbling move is defined as the removal of two pebbles from one edge and placing one on the adjacent edge. The maximal matching cover pebbling number, fmmcp(G), of a graph G, is the minimum number of pebbles that must be placed on E(G), such that after a sequence of pebbling moves the set of edges with pebbles forms a maximal matching regardless of the initial configuration. In this paper, we find the maximal matching cover pebbling number for variants of hypercube.

Hamiltonian path, routing, broadcasting algorithms for connected square network graphs

Connected Square Network Graphs (CSNG) in the study of Selcuk (2022) and Selcuk and Tankul (2022) is reconsidered in this paper. Although (CSNG) is a 2-dimensional mesh structure, the most important feature of this graph is that it is a hypercube variant. For this reason, this study focuses on development algorithms that find solutions to various problems for (CSNG) with the help of hypercube. Firstly, an efficient algorithm that finds the Hamiltonian path is given. Further, two different algorithms that perform the mapping of labels in graph and the unicast routing are given. Furthermore, the parallel process for mapping and unicast routing is discussed. Finally, guidelines are given for broadcasting algorithms.

Super Spanning Connectivity of the Folded Divide-and-SwapCube

A k*-container of a graph G is a set of k disjoint paths between any pair of nodes whose union covers all nodes of G. The spanning connectivity of G, κ*(G), is the largest k, such that there exists a j*-container between any pair of nodes of G for all 1≤j≤k. If κ*(G)=κ(G), then G is super spanning connected. Spanning connectivity is an important property to measure the fault tolerance of an interconnection network. The divide-and-swap cube DSCn is a newly proposed hypercube variant, which reduces the network cost from O(n2) to O(nlog2n) compared with the hypercube and other hypercube variants. The folded divide-and-swap cube FDSCn is proposed based on DSCn to reduce the diameter of DSCn. Both DSCn and FDSCn possess many better properties than hypercubes. In this paper, we investigate the super spanning connectivity of FDSCn where n=2d and d≥1. We show that κ*(FDSCn)=κ(FDSCn)=d+2, which means there exists an m-DPC(node-disjoint path cover) between any pair of nodes in FDSCn for all 1≤m≤d+2.

Embedding Augmented Cube into Certain Trees and Windmill Graphs

The technique used in studying the computational capabilities of interconnection networks and task distribution is graph embedding. Based on the recursively constructed graphs, the hypercube network is popular for its structure. Many variants of hypercube are considered in the literature. Augmented cube is considered as one of the best variants of hypercube as it holds many desirable properties like optimal routing in linear time complexity, vertex symmetricity, wide diameter and maximum connectivity. Our work deals with the exact wirelength, when augmented cube is embedded into certain tree and windmill structures.

Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension

In network theory, distance parameters are crucial in analyzing structural aspects of the networks under investigation, including their symmetry, connectedness, and tendency to form clusters. To this end, the metric dimension and the fault-tolerant metric dimension are important distance invariants of networks. In this article, we consider fractal cubic networks, a variant of hypercubes. We first correct their definition from the seminal paper [Engineering Science and Technology, an International Journal 18 (2015) 32–41]. After that, we determine their metric dimension and fault-tolerant metric dimension, which is in striking contrast to the situation with hypercubes, where these invariants are intrinsically difficult.

Many-to-many edge-disjoint paths in (n,k)-enhanced hypercube under three link-faulty hypotheses

One of the most central issues in interconnection networks of parallel and distributed systems is finding edge-disjoint paths that transmit information. Finding as many as possible many-to-many edge-disjoint paths is conducive to improving the fault-tolerance of such networks. As an interconnection network topology, (n,k)-enhanced hypercube Qn,k (1≤k≤n−1), is a momentous variant of well-known hypercube. For integers 1≤l≤n−1 and n≥2, let δ=0 if 1≤l≤n−k, and δ=1 if n−k+1≤l≤n−1. This paper offers a unified method to determine the minimum cardinalities of faulty links in Qn,k, whose malfunction divides this network into several connected components such that each processor has at least l+δ neighbors, each component contains no less than 2l processors and the number of average neighbors for all processors is at least l+δ, respectively. Under these three different link-faulty assumptions, but for the condition of k=2 and l=n−2, the minimum cardinalities of such faulty links share the same value (n−l−δ+1)2l. And the value in the exceptional case is (n−l−δ)2l+1=2n−1. In other words, we find the maximum numbers of many-to-many edge-disjoint paths of Qn,k under the above three hypotheses, which offers refined measurements for the reliability and fault-tolerance of interconnection networks.

Strong Menger Connectivity of Folded Hypercubes with Faulty Subcube

Menger-type problems in interconnection networks have received many attentions in recent years. A connected graph [Formula: see text] is strong Menger (edge) connected if there are [Formula: see text] vertex (edge)-disjoint paths joining any two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. Fault tolerance is an important criterion in the design of interconnection networks. The folded hypercube [Formula: see text] is an important variant of hypercube [Formula: see text] which remains many desirable properties of hypercube. We consider the strong Menger connectivity of folded hypercubes when part of the network is faulty. We show that [Formula: see text] [Formula: see text] is strong Menger (edge) connected. Which means that when a subcube [Formula: see text] is faulty, the surviving graph [Formula: see text] is strong Menger (edge) connected. This generalizes the result of [Formula: see text] in [J. Parallel Distrib. Comput. 138 (2020) 190–198].

Star-structure connectivity of folded hypercubes and augmented cubes

The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $\kappa(G; H)$ (resp. $H$-substructure connectivity $\kappa^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ_{n}$ are attractive interconnected network prototypes for multiple processor systems. In this paper, we obtain that $\kappa(FQ_{n};K_{1,m})=\kappa^{s}(FQ_{n};K_{1,m})=\lceil\frac{n+1}{2}\rceil$ for $2\leqslant m\leqslant n-1$, $n\geqslant 7$, and $\kappa(AQ_{n};K_{1,m})=\kappa^{s}(AQ_{n};K_{1,m})=\lceil\frac{n-1}{2}\rceil$ for $4\leqslant m\leqslant \frac{3n-15}{4}$.

Fault-tolerability of the hypercube and variants with faulty subcubes

A faulty vertex may probably affect its neighbors and further causes them being faulty, which makes a subnetwork (structure) fail. Therefore, looking into the effect caused by some structures becoming faulty is meaningful. The connectivity and diagnosability are two important parameters to evaluate the fault-tolerability of networks. The connectivity of a network is the minimum number of vertices whose removal will disconnect the network or trivial. We call the network to be t-diagnosable if the number of faulty vertices does not exceed t and all faulty vertices can be identified without a replacement. Let Qn be the n-dimensional hypercube and EH(s,t) be the exchanged hypercube, which is the variant of hypercube. In this paper, we study connectivity, diagnosability, and 1-good-neighbor conditional diagnosability based on structure faults, respectively. Specifically, we first determine the connectivity (1≤k≤n−1 and n≥3) and diagnosability (1≤k≤n−1 and n≥4) of Qn−Qk under the PMC model. Then, we determine the connectivity (2≤k≤min⁡{s,t}) and diagnosability (2≤k≤min⁡{s,t} and min⁡{s,t}≥3) of EH(s,t)−Qk under the PMC model. Finally, we show that the 1-good-neighbor conditional diagnosability of Qn−Qk is 2n−3 for n≥5 and 1≤k≤n−1 under the PMC model, which is almost twice as the traditional diagnosability for a large n.

Reliability Evaluation of Bicube-Based Multiprocessor System under the g-Good-Neighbor Restriction

Multiprocessor systems are commonly deployed for big data analysis because of evolution in technologies such as cloud computing, IoT, social network and so on. Reliability evaluation is of significant importance for maintenance and improvement of fault tolerance for multiprocessor systems, and system-level diagnosis is a primary strategy to identify the faulty processors in the systems. In this paper, we first determine the [Formula: see text]-good-neighbor connectivity of the [Formula: see text]-dimensional Bicube-based multiprocessor system [Formula: see text], a novel variant of hypercube. Besides, we establish the [Formula: see text]-good-neighbor diagnosability of the Bicube-based multiprocessor system [Formula: see text] under the PMC and MM* models.

Structure and substructure connectivity of divide-and-swap cube

High fault tolerance and reliability of multiprocessor systems, modeled by interconnection network, are of great significance to assess the flexibility and effectiveness of the systems. Connectivity is an important metric to evaluate the fault tolerance and reliability of interconnection networks. As classical connectivity is not suitable for such large scale systems, a novel and generalized connectivity, structure connectivity and substructure connectivity, has been proposed to measure the robustness of networks and has witnessed rich achievements. The divide-and-swap cube DSCn is an interesting variant of hypercube that has nice hierarchical properties. In this paper, we mainly investigate H-structure-connectivity, denoted by κ(DSCn;H), and H-substructure-connectivity, denoted by κs(DSCn;H), for H∈{K1,K1,1,K1,m(2≤m≤d+1),C4}, respectively. In detail, we show that κ(DSCn;K1)=κs(DSCn;K1)=d+1 for n≥2, κ(DSCn;K1,1)=κs(DSCn;K1,1)=d+1 for n≥8, κ(DSCn;K1,m)=κs(DSCn;K1,m)=⌊d2⌋+1 with 2≤m≤d+1 for n≥4, κ(DSCn;C4)=3+2(d−2) for 4≤n≤8, ⌊d2⌋+1≤κ(DSCn;C4)≤d+1 for n≥16 and κs(DSCn;C4)=⌊d2⌋+1 for n≥4. Finally, we compare and analyze the ratios of structure (resp. substructure) connectivity to vertex degree of divide-and-swap cube with that of several well-known variants of hypercube.

Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

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].

Minimum layout of hypercubes and folded hypercubes into the prism over caterpillars

The problem of embedding an n-node graph G into an n-node graph H is an important aspect in parallel and distributed processing. Graph embedding results have been successfully used to establish equivalence of interconnections in parallel and distributed machines. The binary hypercube is one of the most widely used multiprocessor systems due to its simple, deadlock-free routing and broadcasting properties. The folded hypercube is an important variant of hypercube with the same number of nodes. Trees are the fundamental graph theoretical models in many fields including artificial intelligence and various network designs. In this paper, we consider the problem of embedding the hypercube and folded hypercube into the prism over a caterpillar in such a way as to minimise its layout.