Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc.

In this paper, we describe a scalable parallel algorithm for sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1024 processors on a Cray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithm substantially improves the state of the art in parallel direct solution of sparse linear systems—both in terms of scalability and overall performance. It is a well known fact that dense matrix factorization scales well and can be implemented efficiently on parallel computers. In this paper, we present the first algorithm to factor a wide class of sparse matrices (including those arising from two- and three-dimensional finite element problems) that is asymptotically as scalable as dense matrix factorization algorithms on a variety of parallel architectures. Our algorithm incurs less communication overhead and is more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems and for Gaussian elimination of diagonally dominant matrices that are almost symmetric in structure. An implementation of our sparse Cholesky factorization algorithm delivers up to 20 GFlops on a Cray T3D for medium-size structural engineering and linear programming problems. To the best of our knowledge,

this paper is organized as follows: Section 2 briefly describes the various ideas and algorithms implemented in METIS. Section 3 describes the user interface to the METIS graph partitioning and sparse matrix ordering packages. Sections 4 and 5 describe the formats of the input and output files used by METIS. Section 6 describes the stand-alone library that implements the various algorithms implemented in METIS. Section 7 describes the system requirements for the METIS package. Appendix A describes and compares various graph partitioning algorithms that are extensively used.

Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity, and provide excellent graph partitions. However, there exists little theoretical analysis that could explain the ability of multilevel algorithms to produce good partitions. In this paper we present such an analysis. We show under certain reasonable assumptions that even if no refinement is used in the uncoarsening phase, a good bisection of the coarser graph is worse than a good bisection of the finer graph by at most a small factor. We also show that the size of a good vertex-separator of the coarse graph projected to the finer graph (without performing refinement in the uncoarsening phase) is higher than the size of a good vertexseparator of the finer graph by at most a small factor.

Contents 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2 Modeling Mesh-based Computations as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.3 Static Graph Partitioning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3.1 Geometric Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.3.2 Combinatorial Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0.3.3 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.3.4 Multilevel Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0.3.5 Combined Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 0.3.6 Qualitative Comparison of Graph Partitioning Schemes . . . . . . . . . . . . . . . . . 16 0.4 Load Balancing of Adaptive Computations . . . . . .

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This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performance machines. In the first part we discuss optimizations of a sequential algorithm to exploit the memory hierarchies that exist in most RISC-based superscalar computers. We begin with the left-looking supernode-column algorithm by Eisenstat, Gilbert and Liu, which includes Eisenstat and Liu's symmetric structural reduction for fast symbolic factorization. Our key contribution is to develop both numeric and symbolic schemes to perform supernodepanel updates to achieve better data reuse in cache and floating-point register...

Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...

The main cost of solving a linear programming problem using an interior point method is usually the cost of solving a series of sparse, symmetric linear systems of equations, A\ThetaA T x = b. These systems are typically solved using a sparse direct method. The first step in such a method is a reordering of the rows and columns of the matrix to reduce fill in the factor and/or reduce the required work. This paper evaluates several methods for performing fill-reducing ordering on a variety of large-scale linear programming problems. We find that a new method, based on the nested dissection heuristic, provides significantly better orderings than the most commonly used ordering method, minimum degree. 1 Introduction An interior point method solves a linear programming problem by computing a sequence of direction vectors. At each iteration, the method takes a step in the computed direction, moving closer to the optimal solution. The details of the interior point method are not relevant ...

When performing sparse matrix factorization, the ordering of matrix rows and columns has a dramatic impact on the factorization time. This paper describes an approach to the reordering problem that produces significantly better orderings than prior methods. The algorithm is a hybrid of nested dissection and minimum degree ordering, and combines an assortment of different algorithmic advances. New or improved algorithms are described for graph compression, multilevel partitioning, and separator improvement. When these techniques are combined, the resulting orderings average 39% better than minimum degree over a suite of test matrices, while requiring roughly 2.7 times the run time of Liu's multiple minimum degree.