Reordering sparse matrices for parallel elimination

doi:10.1016/0167-8191(89)90064-1

Parallel Computing

Volume 11, Issue 1, July 1989, Pages 73-91

https://doi.org/10.1016/0167-8191(89)90064-1 Get rights and content

Abstract

We consider the problem of finding equivalent reorderings of a sparse matrix so that the reordered matrix is suitable for parallel Gaussian elimination. The elimination tree structure is used as our parallel model. We show that the reordering scheme by Jess and Kees generates an elimination tree with minimum height among all such trees from the class of equivalent reorderings. A new height-reducing algorithm based on elimination tree rotation is also introduced. Experimental results are provided to compare these two approaches. The new reordering algorithm using rotation is shown to produce trees with minimum or near-minimum height. Yet, it requires significantly less reordering time.

References (23)

I.S Duff
Parallel implementation of multifrontal schemes
Parallel Comput.
(1986)
J.W.H Liu
Computational models and task scheduling for parallel sparse Cholesky factorization
Parallel Comput.
(1986)
F.J Peters
Parallel pivoting algorithms for sparse symmetric matrices
Parallel Comput.
(1984)
D.J Rose
A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations
A.V Aho et al.
Data Structures and Algorithms
(1983)
G Alaghband et al.
Multiprocessor sparse L/U decomposition with controlled fill-in
R Betancourt
Optimal ordering of sparse matrices for parallel trianulation
D Calahan
Parallel solution of sparse simultaneous linear equations
E.G Coffman
Computer and Job Shop Scheduling Theory
(1976)
I.S Duff et al.
Sparse matrix test problems
SIGNUM Newslett.
(June 1982)

J.A George et al.

Sparse Cholesky factorization on a local-memory multiprocessor

SIAM J. Sci. Statist. Comput.

(1988)

Cited by (38)

Solution of Linear Systems
2005, Handbook of Numerical Analysis
Citation Excerpt :
These elimination trees exhibit little concurrency so that a subtree mapping leads to significant load imbalances. However, Liu [1988, 1989] considered a method that preserves the operation count and the fill-in of the graph of G(PAPT) and at the same time it is more appropriate for parallel elimination. Due to the degree approach, the minimum degree is a greedy algorithm.
This chapter gives an overview over different sparse direct methods and the related literature. The chapter discusses state of art numerical methods for the solution of linear systems. These methods fall belong to the traditional two different classes: direct solution methods based on Gaussian elimination techniques and iterative methods. Developing an efficient parallel or even serial, direct solver for sparse systems of linear equations is a challenging task that has been a subject of research for the past four decades. Recent algorithmic improvements alone have reduced the time required for the sequential direct solution of unsymmetric sparse systems of linear equations by almost an order of magnitude. The main conclusions that can be drawn from this chapter are: (1) the multilevel nested dissection algorithm used to find a fill-in reducing ordering is substantially better than multiple minimum degrees for three-dimensional (3D) irregular matrices, and (2) the multiple minimum degree method performs better for most of the two-dimensional (2D) problems.
The impact of high-performance computing in the solution of linear systems: Trends and problems
2000, Journal of Computational and Applied Mathematics
Citation Excerpt :
A problem with the minimum degree ordering is that it tends to give elimination trees that are not well balanced and so not ideal for use as a computational graph for driving a parallel algorithm. The elimination tree can be massaged [71] so that it is more suitable for parallel computation but the effect of this is fairly limited for general matrices. The beauty of dissection orderings is that they take a global view of the problem; their difficulty until recently has been the problem of extending them to unstructured problems.
Parallel multi-p method
2000, Computers and Mathematics with Applications
A parallel implementation of the multi-p method is discussed, using the master/slave model and the Parallel Virtual Machine (PVM) message passing library. In a series of performance tests, significant speed-up was achieved in those typical cases for the p-version where there was sufficient computational granularity to justify use of the parallel method. These tests indicate that the algorithms devised for load distribution and load balancing are sufficiently robust. These tests also indicate that, even though communication overhead in a network environment is relatively high, there is significant potential for scaling the method to larger processor ensembles.
Gaussian elimination for the solution of linear systems of equations
2000, Handbook of Numerical Analysis
This chapter discusses Gaussian elimination for the solution of linear systems of equations. It focuses on the problem of obtaining the numerical solution of a linear system on a computer. There are two main classes of algorithms to obtain a solution to equation illustrated in the chapter: iterative methods and direct methods. Iterative methods define a sequence of approximations that are expected to be closer and closer to the true solution in some given norm, stopping the iterations by using some predefined criterion and obtaining a vector that is only an approximation of the solution. Direct methods try to compute the solution doing some combinations and modifications of the equations and after a finite number of floating point operations. As computer floating point operations are only done with a certain precision, the computed solution is generally different from the exact solution even with a direct method. The most used direct methods for general matrices belong to a class collectively known as “Gaussian elimination.”
Developments and trends in the parallel solution of linear systems
1999, Parallel Computing
In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field.
Minimum communication cost reordering for parallel sparse Cholesky factorization
1999, Parallel Computing
In this paper, we consider the problem of reducing the communication cost for the parallel factorization of a sparse symmetric positive definite matrix on a distributed-memory multiprocessor. We define a parallel communication cost function and show that, with a contrived example, simply minimizing the height of the elimination tree is ineffective for exploiting minimum communication cost and the discrepancy may grow infinitely. We propose an algorithm to find an ordering such that the communication cost to complete the parallel Cholesky factorization is minimum among all equivalent reorderings. Our algorithm consumes O(nlogn+m) in time, where n is the number of nodes and m the sum of all maximal clique sizes in the filled graph.

View all citing articles on Scopus

^☆: Research was supported in part by the Canadian Natural Sciences and Engineering Research Council under grant A5509, by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems Inc., and by the U.S. Air Force Office of Scientific Research under contract AFOSR-ISSA-86-00012.

View full text

Reordering sparse matrices for parallel elimination☆

Abstract

Parallel Comput.

Parallel Comput.

Parallel Comput.

Data Structures and Algorithms

Multiprocessor sparse L/U decomposition with controlled fill-in

Optimal ordering of sparse matrices for parallel trianulation

Parallel solution of sparse simultaneous linear equations

Computer and Job Shop Scheduling Theory

Sparse matrix test problems

SIGNUM Newslett.

Sparse Cholesky factorization on a local-memory multiprocessor

SIAM J. Sci. Statist. Comput.