Optimization of Wirelength for Embedding Half Hypercubes into Necklace Graphs

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs.

An interconnection network is a programmable system that serves to transport data or messages amongst network components and/or terminals. A network’s topology is typically modeled by a graph. A path of order k in a graph G is a sequence of k distinct vertices, denoted by \(P_k = \langle v_1,v_2,\cdots ,v_k\rangle \), in which any two consecutive vertices are adjacent. The connectivity is a classic index to assess the level of network reliability and fault tolerance. For \(k \ge 2\), a set F of vertex subsets of G is a \(P_k\)-cut if \(G-F\) is disconnected, and each element of F happens to induce a \(P_k\)-subgraph in G. A connected graph G is super \(P_k\)-connected if the smallest component of \(G-F\) is a singleton for every minimum \(P_k\)-cut F of G. A network with smaller diameter can reduce its communication delay in a worst-case perspective. The crossed cube \(CQ_n\) is a hypercube variant whose diameter is about one half of that of the hypercube. This paper is inspired to discover whether \(CQ_n\) is super \(P_k\)-connected for \(k=2,3,4\).

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.

Embedding of hypercubes into necklace, windmill and snake graphs

Since reducing the diameter is likely to improve the performance of an interconnection network, the problem of designing interconnection network with low diameter is still a current research topic. Another important issue in the design of interconnection networks for massively parallel computers is scalability. A new hierarchical interconnection network topology, called Rectangular Twisted Torus Meshes (RTTM network), is proposed. At the lowest level of RTTM network, the Level-1 sub-network, also called a Basic Module, consists of a mesh connection of 2m×2m nodes. Successively higher level networks are built by recursively interconnecting a×2a next lower level sub-networks in the form of a Rectangular Twisted Torus. An appealing property of the RTTM network is its smaller diameter and shorter average distance, which implies a reduction in communication delays. The RTTM network allows the exploitation of computational locality as well as easy expansion up to a million processors.

Load balancing problem is one of the hot issues in cloud computing. Based on Fujian meteorological cloud platform, the establishment of the model carries out quantification the load status and cluster resource utilization rate of virtual machine, and the improved LC algorithm is proposed with the meteorological data features. The verification shows that this algorithm can improve the load balance of resource utilization and cloud computing.

The importance of the large number of thin-diameter and unmyelinated axons that connect different cortical areas is unknown. The pronounced propagation delays in these axons may prevent synchronization of cortical networks and therefore hinder efficient information integration and processing. Yet, such global information integration across cortical areas is vital for higher cognitive function. We hypothesized that delays in communication between cortical areas can disrupt synchronization and therefore enhance the set of activity trajectories and computations interconnected networks can perform. To evaluate this hypothesis, we studied the effect of long-range cortical projections with propagation delays in interconnected large-scale cortical networks that exhibited spontaneous rhythmic activity. Long-range connections with delays caused the emergence of metastable, spatio-temporally distinct activity states between which the networks spontaneously transitioned. Interestingly, the observed activity patterns correspond to macroscopic network dynamics such as globally synchronized activity, propagating wave fronts, and spiral waves that have been previously observed in neurophysiological recordings from humans and animal models. Transient perturbations with simulated transcranial alternating current stimulation (tACS) confirmed the multistability of the interconnected networks by switching the networks between these metastable states. Our model thus proposes that slower long-range connections enrich the landscape of activity states and represent a parsimonious mechanism for the emergence of multistability in cortical networks. These results further provide a mechanistic link between the known deficits in connectivity and cortical state dynamics in neuropsychiatric illnesses such as schizophrenia and autism, as well as suggest non-invasive brain stimulation as an effective treatment for these illnesses.

On oriented diameter of [formula omitted]-star graphs

Given a graph G and a non-negative integer g, the g-extra edge connectivity of G is the minimum cardinality of a set of edges in G, if it exists, whose deletion connects G and each remaining component will have more than g vertices. The spined cube, introduced by Zhou, et al. [The spined cube: A new hypercube variant with smaller diameter, Information Processing Letters, 111 (2011) 561-567.], is a new hypercube variant with smaller diameter. In this paper, we show that the 1-extra edge connectivity of the ridimensionai spined cube is 2n - 2 for n ≥ 3.

Interconnection networks are more vital in telecommunications because of the significant raise in the demand for high-speed networks as a result of the widespread use of computers and the growth of the internet. The hypercube is a versatile network with outstanding qualities that are important for developing extensively in parallel and distributed systems, which include smaller size diameter, recursive structure, symmetry, regularity, low degree, and scalability. In the realm of distributed systems, scalability is seen as an elasticity component of interconnection networks. Fractal cubic networks, a new and novel variant of hypercubes, were recently investigated and have very important qualities such as scalability and better bisection width than traditional hypercubes. The task of assigning channels can be represented as a graph coloring problem. The vertices of a graph represent the transmitters, and if two transmitters are in close proximity to each other, their corresponding vertices are considered nearby. Wavelength assignment enhances the efficiency of wavelength-routed networks by determining routes and assigning wavelengths to connection requests while adhering to network topology and wavelength constraints. Investigation of the acyclic, acyclic edge, star, and star edge chromatic numbers for this newly proposed interconnection network, which is in striking contrast to the situation with hypercubes, where these invariants are intrinsically difficult. In this paper, we establish that for Fractal Cubic Networks (FCNs), the acyclic chromatic number is for . Additionally, for , the star chromatic number and the acyclic edge chromatic number are both 4, while the star edge chromatic number is .

The hypercube has been widely used as the interconnection network in parallel computers. The crossed cube is an variation of hypercube and preserves many of its desirable properties. The hierarchical crossed cube draws upon constructions used within the hypercube and also the crossed cube and is suitable for massively parallel systems with thousands of processors and owns many alluring features, such as symmetry and logarithmic diameter. Embedding cycles into interconnection networks is an important issue for the design of interconnection networks and cycle covering is a well-studied problem in computer science. In this paper, we propose a scheme for a variant of cycle covering problem in hierarchical crossed cubes which all cycles have the same length and each cycle contains the same number of vertices in each crossed cube. Furthermore, we obtain a lower bound for the number of uniform disjoint cycle covers in hierarchical crossed cubes.

The power of butterfly-like networks as multicomputer interconnection networks is studied, by considering how efficiently the butterfly can emulate other networks. Emulations are studied formally via graph embeddings, so the topic here becomes : How efficiently can one embed the graph underlying a given interconnection network in the graph underlying the butterfly network ? Within this framework, the slowdown incurred by an emulation is measured by the sum of the dilation and the congestion of the corresponding embedding (respectively, the maximum amount that the embedding stretches an edge of the guest graph, and the maximum traffic across any edge of the host graph) ; the efficiency of resource utilization in an emulation is measured by the expansion of the corresponding embedding (the ratio of the sizes of the host to guest graph). Three main results expose a number of optimal emulations by butterfly networks. Call a family of graphs balanced if complete binary trees can be embedded in the family with simultaneous dilation, congestion, and expansion 0(1). (1) The family of butterfly graphs is balanced. (2) (a) Any graph < from a family of maxdegree-d graphs having a recursive separator of size S(x) can be embedded in any balanced graph family with simultaneous dilation O(log(d Σ i S(2 -i |G|))) and expansion O(1). (b) Any dilation-D embedding of a maxdegree-d graph in a butterfly graph can be converted to an embedding having simultaneous dilation O(D) and congestion O(dD). (3) Any embedding of a planar graph G in a butterfly graph must have dilation Ω(log Σ (G)/Φ(G), where : Σ(G) is the size of the smallest (1/3, 2/3)-node-separator of G, and Φ(G) is the size of G's largest interior face. Applications of these results include : (1) The n-node X-tree network can be emulated by the butterfly network with slowdown O(log log n) and expansion 0(1) ; no embedding has dilation smaller than Ω(log log n), independent of expansion. (2) Every embedding of the n x n mesh in the butterfly graph has dilation Ω(log n) ; any expansion-O(1) embedding in the butterfly graph achieves dilation O(log n). These applications provide the first examples of networks that can be embedded more efficiently in hypercubes than in butterflies. We also show that analogues of these results hold for networks that are structurally related to the butterfly network. The upper bounds hold for the hypercube and the de Bruijn networks, possibly with altered constants. The lower bounds hold-at least in weakened form-for the de Bruijn network.

The spined cube: A new hypercube variant with smaller diameter

Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we embed recursive circulants into certain necklace graphs for minimizing the wirelength.

Fault-tolerant basis and fault-tolerant edge basis of three classes of French windmill graphs