Toroidal Mesh Research Articles

Extension of the Ingenious Electrostatic Model to Electromagnetic Model for Large-Scale Plasma Simulation with Self-Consistent Electron Dynamics

An ingenious model for large-scale electromagnetic (EM) plasma simulations is proposed. By introducing a dielectric tensor ε with enlarged permittivity elements ε∗ » ε0 to Poisson equation, ∇· (ε∇φ) = −ρ (ε0 is the permittivity of free space, φ is electrostatic potential and ρ is charge density), the Debye length is artificially elongated and the large-scale system can be numerically treated even for including the self-consistent electron dynamics [T. Takizuka et al., Plasma Fusion Res.13, 1203088 (2018)]. In cylindrical coordinates (R, θ, Z) for three-dimensional tokamak simulations, a toroidal element εθθ is chosen much larger than poloidal elements εRR = εZZ = ε∗, and a toroidal mesh size Δθ can be set much larger than poloidal mesh sizes ΔR,Z. Resultantly the total mesh number becomes reasonably small and computation cost can be reduced. EM responses are also simulated using a modified Darwin model for Ampere's law, ∇2 A = −μ0(J − ε∂∇φ/∂t)(A is magnetic vector potential, J is current density, and μ0 is the permeability of free space), where light-speed EM waves are neglected. This modification is consistent with the charge-density continuity.

Design and Evaluation of a Scalable Engine for 3D-FFT Computation in an FPGA Cluster

The Three Dimensional Fast Fourier Transform (3D-FFT) is commonly used to solve the partial differential equations describing the system evolution in several physical phenomena, such as the motion of viscous fluids described by the Navier-Stokes equations. Simulation of such problems requires the use of a parallel High-Performance Computing architecture since the size of the problem grows with the cube of the FFT size, and the representation of the single point comprises several double precision floating- point complex numbers. Modern High-Performance Computing (HPC) systems are considering the inclusion of FPGAs as components of this computing architecture because they can combine effective hardware acceleration capabilities and dedicated communication facilities. Furthermore, the network topology can be optimized for the specific calculation that the cluster must perform, especially in the case of algorithms limited by the data exchange delay between the processors. In this paper, we explore an HPC design that uses FPGA accelerators to compute the 3DFFT. We devise a scalable FFT engine based on a custom radix-2 double-precision core that is used to implement the Decimation in Frequency version of the Cooley-Tukey FFT algorithm. The FFT engine can be adapted to different technology constraints and networking topologies by adjusting the number of cores and configuration parameters in order to minimize the overall calculation time. We compare the various possible configurations with the technological limits of available hardware. Finally, we evaluate the bandwidth required for continuous FFT execution in the APEnet toroidal mesh network.

Strong rainbow connection numbers of toroidal meshes

In 2011, Li et al. [The (strong) rainbow connection numbers of Cayley graphs on Abelian groups, Comput. Math. Appl. 62(11) (2011) 4082–4088] obtained an upper bound of the strong rainbow connection number of an [Formula: see text]-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.

The total bondage numbers and efficient total dominations of vertex-transitive graphs

The total domination number of a graph G without isolated vertices is the minimum number of vertices that dominate all vertices in G. The total bondage number of G is the minimum number of edges whose removal enlarges the total domination number. In this paper, we establish a tight lower bound for the total bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the relation between the total bondage number and the efficient total domination. As applications, we study the total bondage numbers for some circulant graphs and toroidal meshes by characterizing the existence of efficient total dominating sets in these graphs.

Minimizing cyclic cutwidth of graphs using a memetic algorithm

Cyclic cutwidth minimization problem (CCMP) consists of embedding a graph onto a circle such that the maximum cutwidth in a region is minimized. It is an NP-complete problem and for some classes of graphs, exact results of cyclic cutwidth have been proved in literature. However, no algorithm has been reported for general graphs. In this paper, a memetic algorithm is proposed for CCMP in which we have designed six construction heuristics in order to generate a good initial population and also local search is employed to improve the solutions in each generational phase. The algorithm achieves optimal results for the classes of graphs with known exact results. Extensive experiments have also been conducted on some classes of graphs for which exact results are not known such as the complete split graph, join of hypercubes, toroidal meshes, cone graph and some instances of Harwell-Boeing graphs which are a subset of public domain Matrix Market library. Trends observed in the experimental results for some of the classes of graphs have led to conjectures for cyclic cutwidth.

On minimizing vertex bisection using a memetic algorithm

Vertex Bisection Minimization problem (VBMP) consists of partitioning a vertex set into two sets B and B′ where |B|=|V|/2 such that vertex width, VW, is minimized which is defined as the number of vertices in B which are adjacent to at least one vertex in B′. It is an NP-complete problem in general. VBMP has applications in fault tolerance and is related to the complexity of sending messages to processors in interconnection networks via vertex disjoint paths. In this paper, a memetic algorithm has been designed for this problem (MAVBMP) in which four construction heuristics have been proposed to generate the initial population. These heuristics are analyzed statistically and accordingly used in proportion to generate the initial population for MAVBMP. A new crossover type search operator has been proposed for recombination and a local improvement operator has also been developed. Extensive experiments have been conducted on several classes of graphs such as complete bipartite graphs, 2-dimensional ordinary meshes, 2-dimensional toroidal meshes, 3-dimensional toroidal meshes, complete split graph, join of hypercubes and Harwell-boeing graphs which are a subset of public domain Matrix Market library. Trends observed in the experimental results for some of the classes of graphs have led to conjectures for vertex width.

Exact solutions for Latency-Bounded Target Set Selection Problem on some special families of graphs

In the t-Latency-Bounded Target Set Selection (t-LBTSS) problem, we are given a simple graph G=(V,E), a certain latency bound t and a threshold function θ(v)=⌈ρd(v)⌉ for every vertex v of G, where 0<ρ<1 is a rational number and d(v) is the degree of v in V, the goal is to find a target set S with smallest cardinality such that all vertices in V are activated by S by a so called “diffusion process” within t rounds as follows: Initially, all vertices in the target set become activate. Then at each step i of the process, each vertex get activated if the number of active vertices in its neighbor after i−1 exceeds its threshold.For general graphs, the t-LBTSS problem is not only NP-hard, it is also hard to be approximated by Chen’s inapproachability results (Chen, 2009). In this paper, we are interested in finding an optimal target set for some special family of graphs. A simple, tight but nontrivial inequality was presented which gives the lower bound of the total sum of degrees in a feasible target set to t-LBTSS problem, in terms of the number of edges in the graph. Necessary and sufficient conditions for equality to hold have been established, based on which we are able to construct families of infinite number of graphs for which the optimal solution to t-LBTSS problem become obvious. In particular, we gave an exact formula for the optimal solution of a kind of toroidal mesh graphs, while it seems difficult to tell what the optimal solutions are for these graphs without using the equality given in the paper.

The Kirchhoff Index of Toroidal Meshes and Variant Networks

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the toroidal meshes networkTm×nand its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofL(Tm×n),S(Tm×n),T(Tm×n), andC(Tm×n)were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

L-Networks: A Topological Model for Regular 2D Interconnection Networks

A complete family of Cayley graphs of degree four, denoted as L-networks, is considered in this paper. L-networks are 2D mesh-based topologies with wrap-around connections. L-networks constitute a graph-based model which englobe many previously proposed 2D interconnection networks. Some of them have been extensively used in the industry as the underlying topology for parallel and distributed computers of different scales. Tori, twisted and doubly twisted tori, toroidal diagonal meshes, chordal rings, and circulant graphs are, among others, members of the L-network family. Therefore, many results obtained in previous studies on these networks can be deduced from the general framework presented in this work. In addition, the network model presented in this work allows for new results on the domain of low-degree interconnection networks. Particularly, closed expressions for the graph distance properties have been derived and an optimal routing algorithm of constant complexity is provided. Since symmetry has a big impact on network performance, we have also identified which L-networks are symmetric by studying their group of automorphisms. Finally, a very simple model that predicts the performance of L-networks is also presented. Such model has been contrasted with empirical evaluation.

$$K$$ -page crossing number minimization problem: An evaluation of heuristics and its solution using GESAKP

The \(K\)-page crossing number minimization problem (KPMP) is to determine the minimum number of edge crossings over all \(K\)-page book drawings of a graph \(G\) with the vertices placed in a sequence along the spine and the edges on the \(K\)-pages of the book. In this paper we have (a) statistically evaluated five heuristics for ordering vertices on the spine for minimum number of edge crossings with all the edges placed on a single page, (b) statistically evaluated five heuristics for distributing edges on \(K\)-pages with minimum number of crossings for a fixed ordering of vertices on the spine and (c) implemented and experimentally evaluated an instance of guided evolutionary simulated annealing (GESA) called GESAKP here for solving the KPMP. In accordance with the results of (a) and (b) above, in GESAKP, placement of vertices on the spine is decided using a random depth first search of the graph and an edge embedding heuristic is used to distribute the edges on \(K\)-pages of a book. Extensive experiments have been carried out on a suite of benchmark, standard and random graphs to compare the performance of GESAKP with variants of the simple genetic algorithm and other existing approaches. In order to improve the results for some graphs, simple extensions to GESAKP were made. Experiments show that in almost all cases, zero or low cost could be achieved for \(K\le 5\). Also, for \(K \le \) ‘known upper bound’ i.e. upper bound for minimum number of pages necessary to draw or embed the edges of a graph without crossings, zero crossings were obtained. In general, GESAKP outperformed the other techniques. From our experimental results we also present the conjectures for the \(K\)-page crossing number of some complete tripartite graphs and pagenumber of toroidal meshes and a class of complete bipartite and tripartite graphs.

An inflation algorithm and a toroidal mesh algorithm for edge-bipartite graphs

Abstract We present algorithmic methods used in the spectral analysis of bigraphs Δ by means of the Coxeter matrix Cox Δ , the Coxeter spectrum specc Δ , and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply-laced Dynkin diagram D Δ, and defining a Z -congruence of the symmetric Gram matrices G Δ and G D Δ . We also present a toroidal mesh algorithm that allows us to construct a Z -congruence of the non-symmetric Gram matrices G ˇ Δ and G ˇ Δ ′ , if the Coxeter spectra specc Δ and specc Δ ′ coincide.

Expanded Borel Cayley Graphs (Ex-BCGs): A novel communication topology for multi-agent systems

In our early work, we reported Borel Cayley Graphs (BCGs) have the best topological and graph-spectral properties compared to toroidal and diagonal mesh networks and small-world networks. However, BCGs' dependence on generating parameters in size selection and network characteristics has been a challenge in network applications. In this paper, we propose Expanded Borel Cayley Graphs (Ex-BCGs) as a communication topology for multi-agent systems. Ex-BCGs are group-theoretically expanded graphs from BCGs with fixed generating parameters. Ex-BCGs not only conserve constant nodal degree and node labeling scheme but also inherit the communication efficiency of the original BCGs. With experimental results, we show the proposed Ex-BCGs no longer require new generating parameters for new sizes. Moreover, we show resolution of this constraint to produce efficient topological and graph-spectral properties and fast convergence speed when compared to BCGs of similar sizes. We also present the Cut-Through Rewiring (CTR) algorithm as a fault-tolerant methodology against node-failures by regarding pruning nodes as node-failures and utilizing the CTR as a way of recovering connections for neighbors of the pruned nodes in multi-agent systems. The numerical results show that resized networks from Ex-BCGs by the CTR algorithm are more fault-tolerant with robust connectivity, efficient topological properties and faster convergence speed than BCGs.

Total and paired domination numbers of toroidal meshes

Let G be a graph without isolated vertices. The total domination number of G is the minimum number of vertices that can dominate all vertices in G, and the paired domination number of G is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles C n and C m for any n?3 and m?{3,4}, and gives some upper bounds for n,m?5.

Domination numbers of undirected toroidal mesh

The (d,m)-domination number γd,m is a new measure to characterize the reliability of resources-sharing in fault tolerant networks, in some sense, which can more accurately characterize the reliability of networks than the m-diameter does. In this paper, we study the (d, 4)-domination numbers of undirected toroidal mesh \(C_{d_1 } \times C_{d_2 }\) for some special values of d, obtain that \(\gamma _{d,4} \left( {C_{d_1 } \times C_3 } \right) = 2\) if and only if d4(G) − e1 ≤ d < d4(G) for d1 ≥ 5, \(\gamma _{d,4} \left( {C_{d_1 } \times C_4 } \right) = 2\) if \(d_4 \left( G \right) - \left( {2e_1 - \left\lfloor {\tfrac{{d_1 + e_1 }} {2}} \right\rfloor } \right) \leqslant d < d_4 \left( G \right)\) for d1 ≥ 24 and \(\gamma _{d,4} \left( {C_{d_1 } \times C_{d_2 } } \right) = 2\) if d4(G) − (e1 − 2) ≤ d < d4(G) for d1 = d2 ≥ 14.

Dynamic monopolies with randomized starting configuration

Properties of systems with majority voting rules have been extensively studied. In this work we focus on the randomized case, where the system is initialized by random initial set of seeds. Our main aim is to find an asymptotic estimate for sampling probability, so that the initial set of seeds is (not) almost surely a dynamic monopoly. After presenting some trivial examples, we present extensive results for toroidal mesh and random 4-regular graph under simple majority scenario.

Finding regions of interest on toroidal meshes

Fusion promises to provide clean and safe energy, and a considerable amount of research effort is underway to turn this aspiration intoreality. This work focuses on a building block for analyzing data produced from the simulation of microturbulence in magnetic confinementfusion devices: the task of efficiently extracting regions of interest. Like many other simulations where a large amount of data are produced,the careful study of ``interesting'' parts of the data is critical to gain understanding. In this paper, we present an efficient approach forfinding these regions of interest. Our approach takes full advantage of the underlying mesh structure in magnetic coordinates to produce acompact representation of the mesh points inside the regions and an efficient connected component labeling algorithm for constructingregions from points. This approach scales linearly with the surface area of the regions of interest instead of the volume as shown with bothcomputational complexity analysis and experimental measurements. Furthermore, this new approach is 100s of times faster than a recentlypublished method based on Cartesian coordinates.

NETWORK DECONTAMINATION IN PRESENCE OF LOCAL IMMUNITY

We consider the problem of decontaminating a network infected by a mobile virus. The goal is to perform the task using as small a team of antiviral agents as possible, avoiding recontamination of disinfected areas. In all the existing literature, it is assumed that the immunity level of a decontaminated site is nil; that is, a decontaminated node, in absence of an antiviral agent on site, may be re-contaminated by any infected neighbour. The network decontamination problem is studied here under a new model of immunity to recontamination: we consider the case when a decontaminated vertex, after the cleaning agent has gone, will become recontaminated only if a majority of its neighbours are infected. We study the impact that the presence of local immunity has on the number of antiviral agents necessary to decontaminate the entire network. We establish both lower and upper bounds on the number cleaners in the case of (multidimensional) toroidal meshes, graphs of vertex degree at most three (e.g., cubic graphs, binary trees, etc.), and of tree networks. In all cases the established bounds are tight. All upper-bound proofs are constructive; i.e., we exhibit decontamination protocol achieving the claimed bound. We also analyze the total number of moves performed by the agents, and establish tight bounds in some cases.

Grundy number of graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 <= i <= k, is adjacent to (i-1) vertices colored with each color j, 1 <= j <= i-1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an algorithm to generate all graphs for a given Grundy number.

COMMUNICATION PERFORMANCE OF LAM/MPI AND MPICH ON A LINUX CLUSTER

High performance parallel computers provide computational rates necessary for computer simulations and intensive computing applications. An important part of a parallel computer program is an MPI software library, which implements communication within parallel applications. Several MPI implementations exist, most widely used among them are LAM/MPI and MPICH. This paper presents results of four basic synthetic tests and two real simulations in LAM/MPI and MPICH environments. Tests were made on a computer cluster composed of 17 dual-processor nodes connected by a toroidal mesh. Results show that on the investigated cluster, LAM outperformed MPICH especially by bidirectional ring communication, and that appropriate trimming of communication parameters significantly contributes to the final parallel performance.

Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes

A multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles.