Toroidal Mesh Research Articles

The Degree-Diameter Problem for Several Varieties of Cayley Graphs I: The Abelian Case

We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. In particular, we find the largest Abelian graphs with 2 generators (dimensions) and a given diameter. (The results for 2 generators are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with 3 generators and a given diameter which we conjecture to be as large as possible; for the directed case, we obtain partial results, which lead to unusual lattice coverings of 3-space. We discuss the asymptotic behavior of the problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.

Concerning two conjectures on the set of fixed points of a complete rotation of a Cayley digraph

In 1996, J.C. Bermond, T. Kodate, S. Perennes and N. Marlin conjectured that the set F σ of fixed points of some complete rotation σ of the toroidal mesh TM( p) k is not separating (that is F σ does not disconnect TM( p) k ). They also conjectured that the set F ω of fixed points of any complete rotation ω of any Cayley digraph is not separating. In this paper, we prove the first conjecture and disprove the second one.

Special Issue: A systolic block-Jacobi SVD solver for processor meshes

We design the systolic version of the two-sided block-Jacobi algorithm for the singular value decomposition (SVD) of matrix A∈R m×n , and m, n even. The algorithm involves the class CO of parallel orderings on the two-dimensional toroidal mesh with p processors. The mathematical background is based on the QR decomposition (QRD) of local data matrices and on the triangular Kogbetliantz algorithm (TKA) for local SVDs in the diagonal mesh processors. Subsequent updates of local matrices in the diagonal as well as nondiagonal mesh processors are required. We show that all updates can be realized by orthogonal modified Givens rotations. These rotations can be efficiently pipelined in parallel in the horizontal and vertical rings of processor through the toroidal mesh. Our solution requires, per one mesh processor, systolic processing elements (PEs) and additional delay elements. The time complexity can be estimated as where w is the number of global sweeps in the two-sided block-Jacobi algorithm and Δ is the length of the global synchronization time step. The VLSI area per mesh processor, measured by the number of vertical and horizontal wires required for its construction, can be estimated as and the combined VLSI area–time complexity per mesh processor is The theoretical speedup can be estimated as Using the mesh processors of fixed inner size , even, it is possible to construct the square two-dimensional toroidal mesh and to compute the SVD of matrix A, the size of the which matches the shape of mesh processors, i.e. In this sense, the systolic algorithm is scalable.

Complete Rotations in Cayley Graphs

As it is introduced by Bermond, Pérennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a complete rotation, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to investigate such Cayley graphs. We relate some symmetries of a graph with potential algebraic symmetries appearing in its definition as a Cayley graph on a group.

Rotational Cayley Graphs on Transposition Generated Groups

Abstract As it is introduced by Bermond, Kodate, and Prennes, some Cayley graphs, in cluding most popular models for interconnection networks admit a special automorphism, called complete rotation. Such anautomorphism is often used to derive algorithms or properties of the underlying graph For example, optimal gossiping algorithms can be easily designed, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to characterize among Cayley graphs defined a group generated by transpositions, which admit a complete rotation.

Scheduling a divisible task in a two-dimensional toroidal mesh

In this paper, a problem of scheduling an arbitrarily divisible task is considered. Taking into account both communication delays and computation time we propose a scheduling method which minimizes total execution time. We focus on two dimensional processor networks assuming a circuit-switching routing mechanism. The scheduling method uses a scattering scheme proposed in Peters and Syska (IEEE Trans. Parallel Distributed Systems 7(3) (1996) 246–255) to distribute parts of the task to processors in a minimum time. We show how to model and solve this problem with a set of algebraic equations. A solution of the latter allows one to analyze the performance of the network depending on various actual parameters of the task and the parallel machine. Though the method is defined for a particular architecture and scattering scheme it can be generalized to analyze other architectures of parallel computer systems.

Minimizing Communication Penalty of Triangular Solvers by Runtime Mesh Configuration and Workload Redistribution

In this article, we study the effects of network topology and load balancing on the performance of a new parallel algorithm for solving triangular systems of linear equations on distributed-memory message-passing multiprocessors. The proposed algorithm employs novel runtime data mapping and workload redistribution methods on a communication network which is configured as a toroidal mesh. A fully parameterized theoretical model is used to predict communication behaviors of the proposed algorithm relevant to load balancing, and the analytical performance results correctly determine the optimal dimensions of the toroidal mesh, which vary with the problem size, the number of available processors, and the hardware parameters of the machine. Further enhancement to the proposed algorithm is then achieved through redistributing the arithmetic workload at runtime. Our FORTRAN implementation of the proposed algorithm as well as its enhanced version has been tested on an Intel iPSC/2 hypercube, and the same code is also suitable for executing the algorithm on the iPSC/860 hypercube and the Intel Paragon mesh multiprocessor. The actual timing results support our theoretical findings, and they both confirm the very significant impact a network topology chosen at runtime can have on the computational load distribution, the communication behaviors and the overall performance of parallel algorithms.

Global Communication on Circuit-Switched Toroidal Meshes

In this paper, we investigate the uses of virtual channels and multiple communication ports to improve the performance of global communication algorithms for cycles and multi-dimensional toroidal meshes. We use a linear cost model to compare the performances of the best single-port algorithms for broadcasting, scattering, gossiping, and multi-scattering with algorithms that can use multiple ports simultaneously. We conclude that the use of multiple ports can enhance performance when propagation costs are dominant and virtual channels can reduce the total start-up costs. The two mechanisms interact to produce different types of trade-offs for the different communication patterns.

Geometry and Diameter Bounds of Directed Cayley Graphs of Abelian Groups

Many popular interconnection network topologies, such as hypercubes and toroidal meshes, are based on Cayley graphs of Abelian groups. The symmetry and algebraic structure of these graphs result in many nice physical properties of the network concerning layout, routing algorithms, and load balancing. There has been interest in low-diameter Abelian--Cayley graphs because of their smaller communication delay and reduced congestion. For any fixed number of nodes n, and any fixed out-degree k, we are interested in how small the diameter of directed Cayley graphs of Abelian groups can be and what these low-diameter graphs look like. We give an upper bound of $n \leq \frac{3(d + 3)^3}{25}$ for the size of directed Abelian--Cayley graphs with k = 3 and diameter d, correcting a previously published result by Hsu and Jia [SIAM J. Discrete Math., 7 (1994), pp. 57--71]. Our method is based on translational tiling techniques and is a generalization of Wong and Coppersmith's method for k = 2 [J. Assoc. Comput. Mach., 21 (1974), pp. 392--402]. Moreover, our method works for all Abelian groups, not just the cyclic case. For k = 3 we give computational results for the largest Abelian--Cayley graph as a function of diameter. When n = 84 m3 , for integer m, there is a network with $n = \frac{(d + 3)^3}{11.95}$ whose diameter is approximately three-fourths of that of a three-dimensional toroidal cube.

An efficient technique for implementing an image-compression neural algorithm on concurrent multiprocessor architectures

The paper describes a parallel implementation of a neural algorithm performing vector quantization for very low bit-rate video compression on toroidal-mesh multiprocessor systems. The neural model considered is a plastic version of the Neural Gas algorithm, whose features are suitable for implementations on toroidal mesh topologies. The architecture adopted, and the data-allocation strategy, enhance the method's scaling properties and remarkable efficiency. The parallel approach is supported by a theoretical analysis of the efficiency of the overall structure. Experimental results on a significant testbed and the fit between predicted and measured values confirm the validity of the parallel approach.

Edge-localized mode simulations in divertor tokamaks

The three-dimensional nonlinear evolution of moderate wavelength pressure-driven edge-localized instabilities is calculated in divertor tokamaks. The evolution consists of several stages: linear growth, nonlinear saturation and turbulence, shear flow generation, pressure outflow to the outboard divertor, and finally, outflow to the inboard divertor. The shear flow generation appears to be an important factor in transporting pressure perturbations to the inboard side, and in causing pressure loss to the inboard divertor. The physical model consists of dissipative compressional reduced magnetohydrodynamics which includes the important effect of sound wave propagation. A novel numerical discretization, using a poloidal unstructured mesh and a staggered toroidal mesh, has been implemented on a parallel, distributed memory computer. Nonlinear, three-dimensional numerical computations include a separatrix X-point in the computational domain.

The wide-diameter of then-dimensional toroidal mesh

In graph theory and a study of transmission delay and fault tolerance of networks, the connectivity and the diameter of a graph are very important and they have been studied by many mathematicians. We studied the wide-diameter of a k-regular k-connected graph which is defined by the maximum of the k-distance between two distinct vertices, when the k-distance between x and y is equal to the least number l such that there exists k vertex-disjoint paths between x and y whose lengths are at most l. Because the wide-diameter of any k-regular k-connected graph is greater than its diameter, a k-regular k-connected graph whose wide-diameter is equal to “1 + its diameter” is optimal. For example, it is known that the hypercube is such a graph. We define n-dimensional toroidal mesh C(d1, d2, ………, dn) with vertices {(x1, ………, xn)|0 ≤ xi < di (1 ≤ i ≤ n)}. Each vertex (x1, ………, xn) is adjacent to 2n other vertices: (x1 ± 1, x2, ………, xn), (x1, x2 ± 1, ………, xn), ………, (x1, x2, ………, xn ± 1), where additions are performed modulo di (1 ≤ i ≤ n). This graph is an n-dimensional orthogonal mesh with global edges, which is 2n-connected and 2n-regular. We show that the graph satisfies the above property with respect to its 2n-diameter and its diameter except for some special cases. © 1996 John Wiley & Sons, Inc.

Doing the twist: diagonal meshes are isomorphic to twisted toroidal meshes

We show that a k/spl times/n diagonal mesh is isomorphic to a (n+k)/2/spl times/(n+k)/2-(n-k)/2/spl times/(n-k)/2 twisted toroidal mesh, i.e., a network similar to a standard (n+k)/2/spl times/(n+k)/2 toroidal mesh, but with opposite handed twists of (n-k)/2 in the two directions, which results in a loss of ((n-k)/2)/sup 2/ nodes.

Isomorphic Routing on a Toroidal Mesh

We study a routing problem that arises on SIMD parallel architectures whose communication network forms a toroidal mesh. We assume there exists a set of k message descriptors {(xi, yi) ∣ i = 1, 2, …, k}, where (xi, yi) indicates that the ith message's recipient is offset from its sender by xi hops in one mesh dimension, and yi hops in the other. Every processor has k messages to send, and all processors use the same set of message descriptors. The SIMD constraint implies that at any routing step, every processor is actively routing messages with the same descriptors as any other processor. We call this Isomorphic Routing. Our objective is to find the isomorphic routing schedule with the minimum makespan. We consider a number of variations on the problem, yielding complexity results from O(k) to NP-complete. Most of our results follow after we transform the problem into a scheduling problem, where it is related to other well-known scheduling problems.

XMESH interconnection network for massively parallel computers

An XMESH is proposed as a suitable interconnection network for massively parallel computers, and the performance of the proposed interconnection network is analysed. The XMESH has the same horizontal links as those of the toroidal mesh (TMESH); however, it has diagonally crossed links instead of vertical links. The proposed XMESH shows desirable characteristics as an interconnection network for massively parallel computers as the number of nodes increases, while retaining the structural advantages of the TMESH such as the symmetric structure and constant degree of K = 4. Analytical performance evaluations show that the XMESH has a shorter diameter, a shorter mean internode distance, and a higher message-completion rate than the TMESH or the diagonal mesh (DMESH). To confirm these results, an optimal self-routing algorithm for the proposed topology is developed and is used to simulate the maximum delay, the average delay and the throughput in the presence of contention. In all cases, the XMESH is shown to outperform the TMESH and the DMESH, and can provide an attractive alternative to those networks in implementing massively parallel computers.

A STUDY OF MASSIVELY PARALLEL SIMULATED ANNEALING ALGORITHMS FOR CHROMOSOME RECONSTRUCTION VIA CLONE ORDERING

Ordering clones from a genomic library into physical maps of whole chromosomes presents a central computational problem in genetics. Chromosome reconstruction via clone ordering is shown to be isomorphic to the NP-complete Optimal Linear Ordering problem. Massively parallel algorithms for simulated annealing based on Markov chain distribution are proposed and applied to the problem of chromosome reconstruction via clone ordering. Perturbation methods and problem-specific annealing heuristics arc proposed and described. These algorithms are implemented on a 2048 processor MasPar MP-2 system which is an SIMD 2-D toroidal mesh architecture. Convergence, speedup and scalability characteristics of the various algorithms are analyzed and discussed. Results indicate that for an optimal clone ordering, a single Markov chain of solution states should not be distributed across more than two adjoining processing elements (PE's) on the MasPar MP-2.

Parallel Computing of Physical Maps— A Comparative Study in SIMD and MIMD Parallelism

Ordering clones from a genomic library into physical maps of whole chromosomes presents a central computational problem in genetics. Chromosome reconstruction via clone ordering is usually isomorphic to the NP-complete Optimal Linear Arrangement problem. Parallel SIMD and MIMD algorithms for simulated annealing based on Markov chain distribution are proposed and applied to the problem of chromosome reconstruction via clone ordering. Perturbation methods and problem-specific annealing heuristics are proposed and described. The SIMD algorithms are implemented on a 2048 processor MasPar MP-2 system which is an SIMD 2-D toroidal mesh architecture whereas the MIMD algorithms are implemented on an 8 processor Intel iPSC/860 which is an MIMD hypercube architecture. A comparative analysis of the various SIMD and MIMD algorithms is presented in which the convergence, speedup, and scalability characteristics of the various algorithms are analyzed and discussed. On a fine-grained, massively parallel SIMD architecture with a low synchronization overhead such as the MasPar MP-2, a parallel simulated annealing algorithm based on multiple periodically interacting searches performs the best. For a coarse-grained MIMD architecture with high synchronization overhead such as the Intel iPSC/860, a parallel simulated annealing algorithm based on multiple independent searches yields the best results. In either case, distribution of clonal data across multiple processors is shown to exacerbate the tendency of the parallel simulated annealing algorithm to get trapped in a local optimum.

Diagonal and toroidal mesh networks

Diagonal and toroidal mesh are degree-4 point to point interconnection models suitable for connecting communication elements in parallel computers, particularly multicomputers. The two networks have a similar structure. The toroidal mesh is popular and well-studied whereas the diagonal mesh is relatively new. In this paper, we show that the diagonal mesh has a smaller diameter and a larger bisection width. It also retains advantages such as a simple rectangular structure, wirability and scalability of the toroidal mesh network. An optimal self-routing algorithm is developed for these networks. Using this algorithm and the existing routing algorithm for the toroidal mesh, we simulated and compare the performance of these two networks with N=35/spl times/71=2485, N=49/spl times/99=4851, and N=69/spl times/139=9591 nodes under a constant system with a fixed number of messages. Deflection routing is used to resolve conflicts. The effects of various deflection criteria are also investigated. We show that the diagonal mesh outperforms the toroidal mesh in all cases, and thus provides an attractive alternative to the toroidal mesh network.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A short proof of the dilation of a toroidal mesh in a path

We give a short proof that the dilation of an m × n toroidal mesh in an mn-vertex path equals 2 min[lcub] m, n[lcub], if m ≠ n and 2 n − 1, if m = n.

A performance analysis of toroidal mesh networks

A performance analysis model for regular multicomputer networks is developed which covers both single- or multiple-message accepting PE models and buffered or unbuffered communication switches. The performance of a four-neighbour mesh with ‘wrapped-around’ boundaries as found in most transputer nets is evaluated in terms of the overall bandwidth vs. the message generation rate and ‘span’ radii. We show that the network will not saturate if the message generation rate is less than unity and the message hopcount is less than six.