Torus Networks Research Articles

AVERAGE GEODESIC DISTANCE OF SIERPIŃSKI NETWORKS ON TORUS

In this paper, we investigate the average geodesic distance on Sierpiński torus networks. We construct Sierpiński torus networks based on the classic Sierpiński carpet in an iterative way. By applying finite patterns on integrals, we deduce the exact value of the average geodesic distance of the Sierpiński torus. Furthermore, the asymptotic formula for the average geodesic distance of the corresponding networks can be obtained.

Conditional [formula omitted]-matching preclusion for [formula omitted]-dimensional torus networks

The k-matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has neither perfect k-matchings nor almost perfect k-matchings. For many networks, their optimal k-matching preclusion sets are precisely those edges incident with a single vertex. In this paper, we introduce the concept of conditional k-matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional k-matching preclusion numbers and all possible minimum conditional k-matching preclusion sets for n-dimensional torus networks with n≥3. In addition, we investigate the relationship between all optimal sets for three kinds of (conditional) matching preclusion problems.

Integer k-matching preclusion of some interconnection networks

For a positive integer k, a k-matching of a graph G is a function f: E(G)→{0,1,…,k} such that ∑e∈Γ(v)f(e)≤k for every vertex v of G, where Γ(v) represents the set of all edges incident to v. The (strong) k-matching preclusion number of G, denoted by (smpk(G))mpk(G), is the minimum size of (edges and vertices) edges whose deletion leaves the remaining subgraph that has neither perfect k-matchings nor almost perfect k-matchings. This is a generalization of the concept matching preclusion problem proposed by Brigham et al. In this paper, we consider the k-matching preclusion problem of some interconnection networks, such as the hypercube-like graphs HLn, the augmented cubes AQn and the torus networks Cm1□Cm2□⋯□Cmn, which are variants of the hypercube graphs Qn. Our result on restricted hypercube-like graphs RHLn generalizes the result of twisted cubes TQn by Chang et al. [3].

A Spectrum-Based Approach to Network Analysis Utilizing Laplacian and Signless Laplacian Spectra to Torus Networks

A Spectrum-Based Approach to Network Analysis Utilizing Laplacian and Signless Laplacian Spectra to Torus Networks

Spatiotemporal patterns of reaction–diffusion systems with advection mechanisms on large-scale regular networks

This paper primarily investigates the dynamics behaviour of reaction–diffusion systems with advection mechanisms on network structures. Introducing the advection mechanism from continuous space to network structure, this study uses the Laplacian matrix of a directed network to describe it and sets up a network-organized reaction–diffusion system with directed migration mechanism based on this. Firstly, we derived the Turing instability conditions of the system under certain restrictions. We then focus on the Turing dynamical of the system on a type of composite circular networks and a type of composite torus networks. Furthermore, through numerical methods, we reveal the existence of travelling Turing patterns in the system under various circumstances, including the significant case where the system only has diffusion behaviours but no directed migration. Simultaneously, the result of numerical simulations shows that on the network structures presented in this paper, increasing the global directed migration strength promotes homogeneity among the connected nodes; increasing the difference of directed migration strengths between two groups is conducive to the occurrence of Turing instability.

Acceleration of generalized replica exchange with solute tempering simulations of large biological systems on massively parallel supercomputer.

Generalized replica exchange with solute tempering (gREST) is one of the enhanced sampling algorithms for proteins or other systems with rugged energy landscapes. Unlike the replica-exchange molecular dynamics (REMD) method, solvent temperatures are the same in all replicas, while solute temperatures are different and are exchanged frequently between replicas for exploring various solute structures. Here, we apply the gREST scheme to large biological systems containing over one million atoms using a large number of processors in a supercomputer. First, communication time on a multi-dimensional torus network is reduced by matching each replica to MPI processors optimally. This is applicable not only to gREST but also to other multi-copy algorithms. Second, energy evaluations, which are necessary for the multistate bennet acceptance ratio (MBAR) method for free energy estimations, are performed on-the-fly during the gREST simulations. Using these two advanced schemes, we observed 57.72 ns/day performance in 128-replica gREST calculations with 1.5 million atoms system using 16,384 nodes in Fugaku. These schemes implemented in the latest version of GENESIS software could open new possibilities to answer unresolved questions on large biomolecular complex systems with slow conformational dynamics.

Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers

Many artificially intelligent systems solve complex health- and agriculture-related problems that require great computational power. Such systems are used for tracking medical records, genome sequence analysis, image-based plant disease detection, food supply chain traceability, and photosynthesis simulation. Massively parallel computers (MPCs) are among those used to solve these computation-intensive problems. MPCs comprise a million nodes; connecting such a large number of nodes is a daunting task. Therefore, hierarchical interconnection networks (HINs) have been introduced to solve this problem. A midimew-connected torus network (MTN) is a HIN that has basic modules (BM) as torus networks that are connected hierarchically by midimew links. This paper presents the performance of MTNs in terms of static topological parameters and cost-effectiveness, as measured through simulations. An MTN was compared with other networks, including mesh, torus, TESH, TTN, MMN, and TFBN. The results showed that our MTN had a low diameter with a high bisection width and arc connectivity. In addition, our MTN had a high cost–performance trade-off factor (CPTF), a high cost-effective factor (CEF), low packing density, and moderate message-traffic density with marginally higher costs, as compared to other networks, due to wire complexity. However, our MTN provided better bandwidth with higher static fault tolerance. Therefore, MTNs are suggested for further evaluation of the effective implementation of MPCs.

An Extensive Power and Performance Analysis for High Dimensional Mesh and Torus Interconnection Networks

The next generation parallel computers are the keys to achieve exascale performance, whereas sequential computers have already been saturated. In order to achieve this mighty target of exascale computing, one of the main challenges is the reduction of power consumption along with achieving suitable performance. Energy efficiency is a key feature to ensure the trade-off between the performances over the required power usages. Hence, to focus on those issues, the target of this article is to analyze the performance versus power usage trade-off for the conventional networks like- Mesh and Torus. High degree networks show much better performance than the low degree of networks. However, high degree networks require higher power usage for their high degree of interconnected links. This article showed that with zero load latency, the 3D Torus could show about 57.07% better performance than a 2D Torus. On the other hand, a 2D Mesh network requires about 24.22% less router power usage than the 3D Mesh, & 5D Torus requires about 66.8% higher router power usage than a 3D Torus network.

On the Crossing Number of a Torus Network

On the Crossing Number of a Torus Network

Extra Connectivity and Structure Connectivity of 2-Dimensional Torus Networks

The 2-dimensional torus networks, denoted by [Formula: see text], is a special case of the famous [Formula: see text]-dimensional torus network. The extra connectivity, structure connectivity and sub-structure connectivity are generalized connectivity to measure the fault-tolerance of the networks. In this paper, we show the 1-extra connectivity of [Formula: see text] for [Formula: see text] and [Formula: see text] and 2-extra connectivity of [Formula: see text] for [Formula: see text]. We also determine the [Formula: see text]-structure connectivity and [Formula: see text]-substructure connectivity of 2-dimensional torus network [Formula: see text] ([Formula: see text] and [Formula: see text]) when the structure subgraph [Formula: see text], where [Formula: see text] and [Formula: see text]. These results are sharp since the connectivity of [Formula: see text] ([Formula: see text]) is 4.

Improving system performance in Homogeneous Multicore Systems

Allocation of parallel load in multicore systems has become a challenging task for high performance computing system. There are several parameters to evaluate the performance of a scheduling algorithm such as task imbalance and execution time. This paper proposes a task scheduling approach that targets multiple cores connected through appropriate interconnection network. The proposed approach utilizes the computing resources effectively by assigning the tasks dynamically among different cores of the system in a realistic time. Each task has its own timeline and multiple sequence of tasks are mapped on different cores of the system. In particular, performance is evaluated on n x n Mesh, DMesh, ZMesh and Torus networks. The load imbalance and execution times are considered as metrics to evaluate the performance of the proposed algorithm. Simulation results are obtained and compared with well-known minimum distance scheduling algorithm which shows reduction in execution time while maintaining the load imbalance. An improvement of 20-30% is obtained in load imbalance for considered multicore systems with improved execution time. The simulation study reveals that the proposed algorithm is best suited to take architectural benefits for mesh-based multicore systems.

Fractional strong matching preclusion of some Cartesian product graphs

The fractional strong matching preclusion number of a graph is the minimum number of edges and vertices whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional strong matching preclusion number for the Cartesian product of a graph and a cycle. As an application, the fractional strong matching preclusion number for torus networks is also obtained.

Doughnutie: An efficient and low‐latency cloud data center network architecture

SummaryData center network provides the supporting infrastructure for cloud computing. Torus interconnection network is widely employed in the cloud data center networks. However, its long network diameter and average path length (APL) are prohibitive factors for supporting the delay‐sensitive applications. With the aim of meeting latency, network capacity, fault‐tolerance, scalability, and energy‐efficiency requirements of cloud data center networks, we propose a new data center network architecture, named Doughnutie. Doughnutie is built by adding a number of flyway connections to the base Torus network to ameliorate its performance. Doughnutie does not follow a single principle for establishing all of its flyways and consequently, each part of the network can profit from all local, faraway, and neither‐local‐nor‐faraway types of connection. Doughnutie decreases both network diameter and APL, which in turn leads to higher communication efficiency and lower network latency. Moreover, an exclusively designed loop‐free load‐balanced adaptive routing algorithm, named Doughnuter, is proposed. Employing the Doughnuter routing algorithm results in more efficient utilization of all the Doughnutie network resources. Theoretical analysis and simulation results also prove the superiorities of Doughnutie in throughput, load balancing, fault tolerance, and energy‐efficiency.

Hardware Implementation of Contention Resolution in a 4 × 4 Torus Network with Binary Optimized Routing Algorithm

Hardware Implementation of Contention Resolution in a 4 × 4 Torus Network with Binary Optimized Routing Algorithm

Lea-TN: leader election algorithm considering node and link failures in a torus network

Torus network topology offers many advantages such as higher speed, lower latency, better fairness, and lower energy consumption. For these kinds of benefits, nowadays, it is used to construct many parallel and distributed systems like IBM Blue Gene, IBM Sequoia, Mira, and Sugon TC8600. In parallel and distributed computing, multiple nodes act together to carry out large tasks fast. Hence, coordination is indispensable among these nodes to complete such tasks. A possible coordination method could be to elect a leader among the nodes. Along with coordination, the elected leader node also manages other activities such as task allocation, result aggregation, clock synchronization, proficient resource sharing management, and communication among the system’s nodes. In this paper, we propose a lower bound $$\varOmega (N\log _3 N)$$ of message complexity on a comparison-based leader election for a 2D torus network (where N is the number of nodes in the network). Next, we sketch a new leader election algorithm (Lea-TN) considering both the node and link failures for a 2D torus network. This Lea-TN is a deterministic and robust algorithm that elects a leader for a synchronous distributed system. The algorithm chooses a leader, even when there are some link or node failures in the system. We consider the number of non-faulty links and the subsisting nodes’ failure rate to elect a reliable leader. We introduce new patterns for sending messages that help reduce the number of exchanged messages and the execution time of the election process. The proposed algorithm (Lea-TN) enables a node to identify its link failures during the election also. Further, we simulate the Lea-TN algorithm and compare its performance with that of the well-known existing algorithms.

Conditional fractional matching preclusion of [formula omitted]-dimensional torus networks

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has no fractional perfect matchings. For many networks, their optimal fractional matching preclusion sets are precisely those edges incident with a single vertex. The probability that all failures concentrate around a vertex is often small. To overcome the shortcoming, we consider the concept of conditional fractional matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional fractional matching preclusion numbers and all possible minimum conditional fractional matching preclusion sets for n-dimensional torus networks with n≥3.

Midimew Connected Torus Network for Next Generation Massively Parallel Computer System

Important scientific and engineering applications need high performance computing. This can be provided by massively parallel computer (MPC) systems. A sensitive step of maintaining such systems is the interconnection network used to interconnect the computing nodes. The topology used effects the network costs and performance significantly. Hierarchal interconnection networks (HIN) were introduced having several attractive features including low latency, low cost, and high fault tolerance. This paper proposes a new HIN called Midimew connect Torus Network (MTN) that provides constant node degree, high arc connectivity, high fault tolerance and a reasonable bisection width. Static network performance evaluation for the proposed MTN has been conducted and compared with other networks. The comparison included conventional topologies such as 2D Mesh and 2D Torus and hierarchal ones which are TESH and TTN. The comparison of result shows that with the cost of extra communication links MTN attains higher fault tolerance than that of TESH and mesh networks, and equal to that of TTN, and less than that of torus network. Also, the hierarchical networks such as MTN, TTN, and TESH yield moderate bisection width; and the bisection width of MTN is lower than that of mesh and torus networks. This paper shows the superiority of the proposed MTN by comparative study of static network performance.

Optimization of Harmonious coloring problem in Uniform theta graph and torus Network

Coloring of graphs has been extraordinarily widened areas of investigation. A coloring of a graph can be depicted by a capacity that maps pieces of a graph into some plan of numbers commonly called labels or hues, all together that some property is satisfied. In this article we deal with the harmonious chromatic number of central graph of generalized uniform theta graph and torus network. A coloring c: V → ℕ of the nodes is a harmonious coloring if and only if the coloring is proper and for each and every line (e 1, e 2) ∈ E the line color (c(e 1), c(e 2)) is distinctive that is it emerge solely once.

On Solving the Decycling Problem in a Torus Network

Modern supercomputers are massively parallel systems: they embody thousands of computing nodes and sometimes several millions. The torus topology has proven very popular for the interconnect of these high‐performance systems. Notably, this network topology is employed by the supercomputer ranked number one in the world as of November 2020, the supercomputer Fugaku. Given the high number of compute nodes in such systems, efficient parallel processing is critical to maximise the computing performance. It is well known that cycles harm the parallel processing capacity of systems: for instance, deadlocks and starvations are two notorious issues of parallel computing that are directly linked to the presence of cycles. Hence, network decycling is an important issue, and it has been extensively discussed in the literature. We describe in this paper a decycling algorithm for the 3‐dimensional k‐ary torus topology and compare it with established results, both theoretically and experimentally. (This paper is a revised version of Antoine Bossard (2020)).

Segment Switching: A New Switching Strategy for Optical HPC Networks

Photonics are becoming realistic technologies for implementing interconnection networks in near future Exascale supercomputer systems. Photonics present key features to design high-performance and scalable supercomputer networks, such as higher bandwidth and lower latencies than their electronic supercomputer networks counterparts. Some research work is focused on conventional network topologies built with photonic technologies, with the aim of taking advantage of photonic characteristics. Nevertheless, these approaches fail in that they keep low the network utilization. We looked into this downside and we found that circuit switching was the main performance limitation. In this article we propose a new switching mechanism, called Segment Switching , to address this constraint and improve the network utilization. Segment Switching splits the circuit in segments of the whole path, and uses buffering on selected nodes on the network. Experimental results show that the devised approach significantly outperforms photonic circuit switching in conventional torus and fat tree networks by 70% and 90%, respectively.