Mesh-connected Processor Array Research Articles

A NOTE ON OFF-LINE PERMUTATION ROUTING ON A MESH-CONNECTED PROCESSOR ARRAY

We show how to off-line route any permutation of an n×n mesh-connected processor array in 2.5n steps with at most two packets per processor per time step and in 2.25n+3 steps with at most four packets per processor per time step.

Parallel strong orientation on a mesh connected computer

We present a solution for the following problem: given an undirected bridgeless connected graph G = ( V, E), find an orientation of each edge such that the resulting directed graph is strongly connected. We assume the input graph to be given as an adjacency matrix stored in the processors of a mesh connected processor array such that each processor contains one entry. Our algorithm is a modification of Tarjan and Vishkin's strong orientation method [13] for a CREW PRAM. It takes time O( n) for an n-vertex graph which is asymptotically optimal for the considered computer model.

On the Generalized Schur Decomposition of a Matrix Pencil for Parallel Computation

An algorithm to solve the generalized eigenvalue problem $Ax = \lambda Bx$ for arbitrary complex $n \times n$ matrices A, B is developed. It is a generalization of an unsymmetric Jacobi method of Eberlein. Each iteration step needs only local information about the current matrix, thus admitting efficient parallel implementations on certain parallel architectures. Versions for an $(n/2) \times (n/2)$ square array of mesh-connected processors and a ring of $n/ 2$ processors are discussed. Convergence performance of the algorithm is compared with a recently proposed Jacobi-like method for matrix pencils based on the unsymmetric Jacobi method of Stewart. The numerical experiments suggest that the method presented here is considerably faster for matrix pencils which are not near to normality.

Processor-time optimal parallel algorithms for digitized images on mesh-connected processor arrays

We present processor-time optimal parallel algorithms for several problems onn ×n digitized image arrays, on a mesh-connected array havingp processors and a memory of sizeO(n 2) words. The number of processorsp can vary over the range [1,n 3/2] while providing optimal speedup for these problems. The class of image problems considered here includes labeling the connected components of an image; computing the convex hull, the diameter, and a smallest enclosing box of each component; and computing all closest neighbors. Such problems arise in medium-level vision and require global operations on image pixels. To achieve optimal performance, several efficient data-movement and reduction techniques are developed for the proposed organization.

An acceleration method for computing the generalized eigenvalue problem on a parallel computer

We give a cubic correction step for improving the current eigenvalue algorithms for computing the generalized Schur decomposition of a regular pencil λB− A using a Jacobi-like method. The correction method can be used to speed up the convergence at the end of the Jacobi-like process when the strictly lower triangular elements of the matrix pair ( A, B) have become sufficiently small; it can be implemented in parallel on an n× n square array of mesh-connected processors in O( n) computational time. A quantitative analysis of the convergence and a comparison of the complexity of one Jacobi sweep versus one correction step are presented.

( k− k) Routing on multidimensional mesh-connected arrays

In this paper we study the problem of routing packets on an r-dimensional mesh-connected array of processors. The focus of this paper is on routing with each processor containing exactly k packets, k ⩾ 2, initially and finally (so-called k−k routing). For two-dimensional n × n grids the number of transport steps is at most 5 4 kn + O(kn/f(n)) with a buffer size of O(kf(n)). In the special case of a sequence of k permutation routing problems this step count can be reduced to kn + O( kn/ f( n)). For an r-dimensional grid, r ⩾ 3, with side length n the same technique yields an algorithm with step count (r − 1)(1 + 1/r 2kn + O(n/f(n) 1 (r − 1) ) and buffer rk · f( n). For sequences of permutation routing problems this drops to [k/r] (2r − 2)n + O(kn/f(n) 1 (r−1) ) and a buffer size of O(kf(n)). Furthermore it is shown that splitting large packets into smaller ones has benefits for permutation routing problems. For grids with wrap-around connections these step counts and times generally can be reduced by one-half.

Space and time complexities of balanced sorting on processor arrays

A processor is balanced in carrying out a computation if its computing time equals its I/O time. When the computation bandwidth of a processor is increased, like when multiple processors are incorporated to form an array, the critical question is to what degree the processor's memory must be enlarged in order to alleviate the I/O bottleneck to keep the computation balanced. In this paper, for the sorting problem, we present two balanced algorithms on linearly connected and mesh-connected processor arrays, respectively, and show that they reach the derived lower bounds of memory sizes. We also verify that the time complexities of the algorithms are optimal under their respective hardware constraints.

Fast hough transform on a mesh connected processor array

We present a method on a mesh connected torus processor array that computes the Hough transform in O(n) time

Time lower bounds for sorting on multi-dimensional mesh-connected processor arrays

The problem of sorting on a mesh-connected processor array has been studied. We develop the chain argument in order that we can apply its extended version to derive nontrivial lower bounds for sorting. We show a theorem that gives a relation between computing time for sorting n d items and the number of processors in a certain region of the mesh-connected model. We can numerically calculate the best lower bound obtainable from the theorem

A simple systolic method to find all bridges of an undirected graph

An algorithm to find all bridges of an undirected graph on a mesh-connected processor array is presented. Asymptotically it has the same complexity in terms of time and space as the best previous known algorithm, the one of Atallah and Kosaraju. But it has a significantly simpler structure and therefore, it gains a constant factor of about two in time.

Time lower bounds for parallel sorting on a mesh-connected processor array

We prove that (1+√6/2)n ≈ 2.22n is a time lower bound independent of indexing schemes for sorting n2 items on an n × n mesh-connected processor array. We distinguish between indexing schemes by showing that there exists an indexing scheme which is provably worse than the snake-like row-major indexing for sorting. We also derive lower bounds for various indexing schemes. All these results are obtained by using the chain argument which we provide in this paper.

Two nearly optimal sorting algorithms for mesh-connected processor arrays using shear-sort

The recently discovered Shear-sort algorithm requires log2 n iterations of row and column sorts for ordering n 2 elements on an n × n mesh-connected array of processors. Although the method is extremely simple and practical, it falls short by a factor of log 2 n of the well-known lower bound for sorting on a mesh-connected computer. Recursive application of Shear-sort in a square array leads to the first proposed O( n) algorithm. A slightly more complicated iterative algorithm, also executing in O( n) steps is presented next. Both algorithms are surprisingly simple and are the result of the adoption of an entirely new approach to two-dimensional sorting.

Image computations on meshes with multiple broadcast

The authors consider a mesh-connected processor array with broadcast, in which the processors in each row and in each column can access a shared bus. This computation model is an abstraction of commercially available parallel machines such as the distributed array processor (DAP). The authors show efficient data movement using the buses, which leads to significantly faster solution times for several image problems compared to those on an ordinary mesh-connected array. For several problems, the time performance is comparable to that on the pyramid of corresponding size. Alternate parallel organizations with broadcast feature are also shown. Utilizing the broadcast feature on the arrays can lead to significant improvement of the time performance of many image computations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parallel rule-based fuzzy inference on mesh-connected systolic arrays

The authors describe a scheme for performing fuzzy-reasoning and fuzzy-inference operations in parallel on mesh-connected array processors such as the geometric arithmetic parallel processor (GAPP) and the advanced systolic array processor (ASAP). The reasoning mechanism is a general, fuzzy, rule-based inference network. The array is configured into stages, and the scheme operates in a pipeline configuration, where different sets of data move sequentially between the stages. The authors investigated the tradeoff between space and time complexity of the proposed technique and achieved a fast, space-efficient implementation of the approach. Before presenting the proposed system, they review briefly the concepts of inexact reasoning, fuzzy inference, and mesh-connected fine-grain parallel computer architecture.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal simulations between mesh-connected arrays of processors

Let G and H be two mesh-connected arrays of processors, where G = g 1 , X g 2 X … X g 1 , H = h 1 x h 2 x … x h d , and g 1 … g 1 ≤ h 1 … h d . The problem of simulating G by H is considered and the best possible simulation in terms of the g i 's and h i 's is characterized by giving such a simulation and proving its optimality in the worst-case sense. Also the same bound on the average cost of encoding the edges of G as distinct paths in H is established.

Self-reconfiguring interconnection network for a fault-tolerant mesh-connected array of processors

An interconnection network capable of spontaneously reconfiguring a mesh-connected processor array on detection of faulty processors is presented. Although the reconfiguration process is global in nature, the network control circuitry is localised around each processor and is therefore completely modular. In addition, the structure of the control circuitry is fixed and thus independent of the array size or the number of spare processors.

Parallel 2-D Convolution on a Mesh Connected Array Processor

In this correspondence, a parallel 2-D convolution scheme is presented. The processing structure is a mesh connected array processor consisting of the same number of simple processing elements as the number of pixels in the image. For most windows considered, the number of computation steps required is the same as that of the coefficients of a convolution window. The proposed scheme can be easily extended to convolution windows of arbitrary size and shape. The basic idea of the proposed scheme is to apply the 1-D systolic concept to 2-D convolution on a mesh structure. The computation is carried out along a path called a convolution path in a systolic manner. The efficiency of the scheme is analyzed for windows of various shapes. The ideal convolution path is a Hamiltonian path ending at the center of the window, the length of which is equal to the number of window coefficients. The simple architecture and control strategy make the proposed scheme suitable for VLSI implementation.

A Local Relaxation Method for Solving Elliptic PDEs on Mesh-Connected Arrays

A local relaxation method for solving linear elliptic PDEs with $O(N)$ processors and $O(\sqrt N )$ computation time is proposed. We first examine the implementation of traditional relaxation algorithms for solving elliptic PDEs on mesh-connected processor arrays, which require $O(N)$ processors and $O(N)$ computation time. The disadvantage of these implementations is that the determination of the acceleration factors requires some global communication at each iteration. The high communication cost increases the computation time per iteration significantly. Therefore, a local relaxation scheme is proposed to achieve the acceleration effect with very little global communication in the loading stage. We use a Fourier analysis approach to analyze the local relaxation method and also show its convergence. The convergence rate of the local relaxation method is studied by computer simulation.

Lower bounds for sorting on mesh-connected architectures

Lower bounds for sorting on mesh-connected arrays of processors are presented. For sorting N=n1 n 2...n r elements on an n 1×n2×... ×n r array 2(n 1+...+n r−1)+n r data interchange steps are needed asymptotically. For two dimensions these bounds are asymptotically best possible provided that n 1 and n 2 are powers of 2. In this case the generalized s 2-way merge sort of Thompson and Kung turns out to be asymptotically optimal. The minimal asymptotic bound of 2 √2N interchange steps can be obtained only by sorting algorithms suitable for √N/2×√2N meshes. For r≧3 dimensions an analysis of aspect-ratios also demonstrates that there exist mesh-connected architectures which are better suited for sorting than simple r-dimensional cubes.

On the Schur Decomposition of a Matrix for Parallel Computation

An algorithm to solve the eigenproblem for nonsymmetric matrices on an N × N array of mesh-connected processors, isomorphic to the architecture described by Brent and Luk for symmetric matrices, is presented. This algorithm is a generalization of the classical Jacobi method, and, as such, holds promise for parallel architectures. The rotational parameters for the nonsymmetric case are carefully analyzed; many examples of a working program, simulating the parallel architecture, are given with experimental evidence of quadratic convergence.