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151
A Separator Theorem for Planar Graphs f
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds su ..."
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Cited by 397 (1 self)
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Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.
Bundle adjustment – a modern synthesis
 Vision Algorithms: Theory and Practice, LNCS
, 2000
"... This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics c ..."
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Cited by 386 (12 self)
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This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.
Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 122 (18 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 105 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
SuperLU DIST: A scalable distributedmemory sparse direct solver for unsymmetric linear systems
 ACM Trans. Mathematical Software
, 2003
"... We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and sc ..."
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Cited by 87 (17 self)
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We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on largescale distributed machines.
How Good is Recursive Bisection?
 SIAM J. Sci. Comput
, 1995
"... . The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the opti ..."
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Cited by 84 (4 self)
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. The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NPcomplete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a pway partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as wellshaped finite element and finite difference...
Permuting Sparse Rectangular Matrices into BlockDiagonal Form
 SIAM Journal on Scientific Computing
, 2002
"... We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. W ..."
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Cited by 57 (19 self)
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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using stateoftheart graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
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.
Robust Ordering of Sparse Matrices using Multisection
 Department of Computer Science, York University
, 1996
"... In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree ..."
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Cited by 46 (2 self)
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In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree and generalized nested dissection. Experimental results show that by using multisection, we obtain an ordering which is consistently as good as or better than both for a wide spectrum of sparse problems. 1 Introduction It is well recognized that finding a fillreducing ordering is crucial in the success of the numerical solution of sparse linear systems. For symmetric positivedefinite systems, the minimum degree [38] and the nested dissection [11] orderings are perhaps the most popular ordering schemes. They represent two opposite approaches to the ordering problem. However, they share a common undesirable characteristic. Both schemes produce generally good orderings, but the ordering qua...