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16
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 144 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
A Framework For Solving Vlsi Graph Layout Problems
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1984
"... This paper introduces a new divideandconquer framework for VLSI graph layout. Universally close upper and lower bounds are obtained for important cost functions such as layout area and propagation delay. The framework is also effectively used to design regular and configurable layouts, to assemble ..."
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Cited by 134 (4 self)
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This paper introduces a new divideandconquer framework for VLSI graph layout. Universally close upper and lower bounds are obtained for important cost functions such as layout area and propagation delay. The framework is also effectively used to design regular and configurable layouts, to assemble large networks of processors using restructurable chips, and to configure networks around faulty processors. It is also shown how good graph partitioning heuristics may be used to develop a provably good layout strategy.
Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems
, 1996
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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 40 (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...
Mapping Algorithms and Software Environment for Data Parallel PDE . . .
 JOURNAL OF DISTRIBUTED AND PARALLEL COMPUTING
, 1994
"... We consider computations associated with data parallel iterative solvers used for the numerical solution of Partial Differential Equations (PDEs). The mapping of such computations into load balanced tasks requiring minimum synchronization and communication is a difficult combinatorial optimization p ..."
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Cited by 34 (19 self)
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We consider computations associated with data parallel iterative solvers used for the numerical solution of Partial Differential Equations (PDEs). The mapping of such computations into load balanced tasks requiring minimum synchronization and communication is a difficult combinatorial optimization problem. Its optimal solution is essential for the efficient parallel processing of PDE computations. Determining data mappings that optimize a number of criteria, likeworkload balance, synchronization and local communication, often involves the solution of an NPComplete problem. Although data mapping algorithms have been known for a few years there is lack of qualitative and quantitative comparisons based on the actual performance of the parallel computation. In this paper we present two new data mapping algorithms and evaluate them together with a large number of existing ones using the actual performance of data parallel iterative PDE solvers on the nCUBE II. Comparisons on the performance of data parallel iterative PDE solvers on medium and large scale problems demonstrate that some computationally inexpensive data block partitioning algorithms are as effective as the computationally expensive deterministic optimization algorithms. Also, these comparisons demonstrate that the existing approach in solving the data partitioning problem is inefficient for large scale problems. Finally, a software environment for the solution of the partitioning problem of data parallel iterative solvers is presented.
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...
Predicting Structure In Nonsymmetric Sparse Matrix Factorizations
 GRAPH THEORY AND SPARSE MATRIX COMPUTATION
, 1992
"... Many computations on sparse matrices have a phase that predicts the nonzero structure of the output, followed by a phase that actually performs the numerical computation. We study structure prediction for computations that involve nonsymmetric row and column permutations and nonsymmetric or nonsqu ..."
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Cited by 30 (8 self)
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Many computations on sparse matrices have a phase that predicts the nonzero structure of the output, followed by a phase that actually performs the numerical computation. We study structure prediction for computations that involve nonsymmetric row and column permutations and nonsymmetric or nonsquare matrices. Our tools are bipartite graphs, matchings, and alternating paths. Our main new result concerns LU factorization with partial pivoting. We show that if a square matrix A has the strong Hall property (i.e., is fully indecomposable) then an upper bound due to George and Ng on the nonzero structure of L + U is as tight as possible. To show this, we prove a crucial result about alternating paths in strong Hall graphs. The alternatingpaths theorem seems to be of independent interest: it can also be used to prove related results about structure prediction for QR factorization that are due to Coleman, Edenbrandt, Gilbert, Hare, Johnson, Olesky, Pothen, and van den Driessche.
Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus
 SIAM Journal of Discrete Mathematics
, 1996
"... This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( ..."
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Cited by 21 (4 self)
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This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut off from G a piece of no more than (1 \Gamma ")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor. Keywords: graph separator, graph genus, algorithm, divideandconquer, topological graph theory AMS(MOS) subject classifications: 05C10, 05C85, 68R10 1 Bulgarian Academy of Sci., CICT, G.Bonchev 25A, 1113 Sofia, Bulgaria 2 Department of Comp.Sci.,Rice University, P.O.Box 1892, Houston, Texas 77251, USA 1 Introduction Let S be a class of graphs closed under t...
NestedDissection Orderings For Sparse Lu With Partial Pivoting
 SIAM J. Matrix Anal. Appl
, 2000
"... . We describe the implementation and performance of a novel fillminimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for ..."
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Cited by 18 (5 self)
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. We describe the implementation and performance of a novel fillminimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for LU with partial pivoting, our new method preorders the columns of the matrix (the row permutation is chosen by the pivoting sequence during the numerical factorization). Also like other methods, the column permutation Q that we select is a permutation that minimizes the fill in the Cholesky factor of Q T A T AQ. Unlike existing columnordering techniques, which all rely on minimumdegree heuristics, our new method is based on a nesteddissection ordering of A T A. Our algorithm, however, never computes a representation of A T A, which can be expensive. We only work with a representation of A itself. Our experiments demonstrate that the method is e#cient and that it can reduce fill significantly relative to the best existing methods. The method reduces the LU running time on some very large matrices (tens of millions of nonzeros in the factors) by more than a factor of 2. 1.
Separators in Graphs with Negative and Multiple Vertex Weights
 ALGORITHMICA
, 1992
"... A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show t ..."
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Cited by 9 (1 self)
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A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show that any separator theorem implies various weighted separator theorems. In particular, we show that if the vertices of the graph have realvalued weights, which may be positive or negative, then the graph can be divided exactly in half according to weight. If k unrelated sets of weights are given, the graph can be divided simultaneously by all k sets of weights. These results considerably strengthen earlier results of Gilbert, Lipton, and Tarjan: (1) for k = 1 with the weights restricted to be nonnegative, and (2) for k > 1, nonnegative weights, and simultaneous division within a factor of (1 + ffl) of exactly in half.