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158
Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems
, 1996
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Fast and Effective Algorithms for Graph Partitioning and Sparse Matrix Ordering
 IBM JOURNAL OF RESEARCH AND DEVELOPMENT
, 1996
"... Graph partitioning is a fundamental problem in several scientific and engineering applications. In this paper, we describe heuristics that improve the stateoftheart practical algorithms used in graphpartitioning software in terms of both partitioning speed and quality. An important use of graph ..."
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Cited by 56 (11 self)
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Graph partitioning is a fundamental problem in several scientific and engineering applications. In this paper, we describe heuristics that improve the stateoftheart practical algorithms used in graphpartitioning software in terms of both partitioning speed and quality. An important use of graphpartitioning is in ordering sparse matrices for obtaining direct solutions to sparse systems of linear equations arising in engineering and optimization applications. The experiments reported in this paper show that the use of these heuristics results in a considerable improvement in the quality of sparsematrix orderings over conventional ordering methods, especially for sparse matrices arising in linear programming problems. In addition, our graphpartitioningbased ordering algorithm is more parallelizable than minimumdegreebased ordering algorithms, and it renders the ordered matrix more amenable to parallel factorization.
Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time
 In Proc. 42nd IEEE Annual Symposium on Foundations of Computer Science
, 2001
"... for finding shortest paths in a planar graph with real weights. ..."
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Cited by 54 (0 self)
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for finding shortest paths in a planar graph with real weights.
Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree
, 1995
"... Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum ..."
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Cited by 52 (4 self)
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Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(logn) (minimum front size and treewidth) and O(log^2 n) (pathwidth and minimum elimination tree height) times the optimal values. In addition, we show that unless P = NP there are no absolute approximation algorithms for any of the parameters.
Preconditioning highly indefinite and nonsymmetric matrices
 SIAM J. SCI. COMPUT
, 2000
"... Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditionin ..."
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Cited by 43 (4 self)
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Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131–151] and by Duff and
Graph Partitioning Algorithms With Applications To Scientific Computing
 Parallel Numerical Algorithms
, 1997
"... 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 su ..."
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Cited by 41 (0 self)
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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...
Compact Representations of Separable Graphs
 In Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queri ..."
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Cited by 36 (11 self)
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We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.
Hybridizing Nested Dissection and Halo Approximate Minimum Degree for Efficient Sparse Matrix Ordering
 IN PROCEEDINGS OF IRREGULAR'99, LNCS 1586
, 1999
"... Minimum degree and nested dissection are the two most popular reordering schemes used to reduce llin and operation count when factoring and solving sparse matrices. Most of the stateoftheart ordering packages hybridize these methods by performing incomplete nested dissection and ordering by ..."
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Cited by 34 (15 self)
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Minimum degree and nested dissection are the two most popular reordering schemes used to reduce llin and operation count when factoring and solving sparse matrices. Most of the stateoftheart ordering packages hybridize these methods by performing incomplete nested dissection and ordering by minimum degree the subgraphs associated with the leaves of the separation tree, but most often only loose couplings have been achieved, resulting in poorer performance than could have been expected. This paper presents a tight coupling of the nested dissection and halo approximate minimum degree algorithms, which allows the minimum degree algorithm to use exact degrees on the boundaries of the subgraphs passed to it, and to yield back not only the ordering of the nodes of the subgraph, but also the amalgamated assembly subtrees, for efficient block computations. Experimental results show the performance improvement of this hybridization, both in terms of fillin reduction and increa...
Spectral Nested Dissection
, 1992
"... . We describe a spectral nested dissection algorithm for computing orderings appropriate for parallel factorization of sparse, symmetric matrices. The algorithm makes use of spectral properties of the Laplacian matrix associated with the given matrix to compute separators. We evaluate the quality of ..."
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Cited by 30 (5 self)
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. We describe a spectral nested dissection algorithm for computing orderings appropriate for parallel factorization of sparse, symmetric matrices. The algorithm makes use of spectral properties of the Laplacian matrix associated with the given matrix to compute separators. We evaluate the quality of the spectral orderings with respect to several measures: fill, elimination tree height, height and weight balances of elimination trees, and clique tree heights. Spectral orderings compare quite favorably with commonly used orderings, outperforming them by a wide margin for some of these measures. These results are confirmed by computing a multifrontal numerical factorization with the different orderings on a Cray YMP with eight processors. Keywords. graph partitioning, graph spectra, Laplacian matrix, ordering algorithms, parallel orderings, parallel sparse Cholesky factorization, sparse matrix, vertex separator AMS(MOS) subject classifications. 65F50, 65F05, 65F15, 68R10 1. Introducti...
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.