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104
A fast and high quality multilevel scheme for partitioning irregular graphs
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1998
"... 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. ..."
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Cited by 803 (12 self)
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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.
METIS  Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 2.0
, 1995
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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. Becaus ..."
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Cited by 86 (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...
Separators for spherepackings and nearest neighbor graphs
 J. ACM
, 1997
"... Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the ..."
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Cited by 72 (7 self)
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Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every knearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a diskpacking, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.
Graph partitioning for high performance scientific simulations. Computing Reviews 45(2
, 2004
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Approximating center points with iterative Radon points
 International Journal of Computational Geometry & Applications
, 1996
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Relationshipbased Clustering and Cluster Ensembles for Highdimensional Data Mining
, 2002
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On The Quality Of Spectral Separators
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 1998
"... Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral met ..."
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Cited by 42 (3 self)
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Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods "work well in practice." We present an initial attempt at such analysis. In particular, we consider two popular spectral separator algorithms, and provide counterexamples that show these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Further, using the second smallest eigenvector of these graphs produces partitions that are poor with respect to bounds on the gap between the isoperimetric number and the cut quotient of the spectral separator. Even if a generalized spectral algorithm uses n # for 0 < # < 1 4 eigenvectors, there exist graphs for which the algorithm fails to find a separator with a cut quotient within n 1 4 #  1 of the isoperimetric number. We also introduce some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices; these facts provide the basis for the analysis of the counterexamples. Finally, we discuss some developments in spectral partitioning that have occurred since these results first appeared.
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...
Dynamic Load Balancing in Computational Mechanics
 Computer Methods in Applied Mechanics and Engineering
"... . In many important computational mechanics applications, the computation adapts dynamically during the simulation. Examples include adaptive mesh refinement, particle simulations and transient dynamics calculations. When running these kinds of simulations on a parallel computer, the work must be a ..."
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Cited by 34 (2 self)
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. In many important computational mechanics applications, the computation adapts dynamically during the simulation. Examples include adaptive mesh refinement, particle simulations and transient dynamics calculations. When running these kinds of simulations on a parallel computer, the work must be assigned to processors in a dynamic fashion to keep the computational load balanced. A number of approaches have been proposed for this dynamic load balancing problem. This paper reviews the major classes of algorithms, and discusses their relative merits on problems from computational mechanics. Shortcomings in the stateoftheart are identified and suggestions are made for future research directions. Key words. dynamic load balancing, parallel computer, adaptive mesh refinement 1. Introduction. The efficient use of a parallel computer requires two, often competing, objectives to be achieved. First, the processors must be kept busy doing useful work. And second, the amount of interprocess...