Results 1  10
of
60
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. ..."
Abstract

Cited by 797 (12 self)
 Add to MetaCart
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.
Multiclass spectral clustering
 In Proc. Int. Conf. Computer Vision
, 2003
"... We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We t ..."
Abstract

Cited by 181 (7 self)
 Add to MetaCart
We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We then solve an optimal discretization problem, which seeks a discrete solution closest to the continuous optima. The discretization is efficiently computed in an iterative fashion using singular value decomposition and nonmaximum suppression. The resulting discrete solutions are nearly globaloptimal. Our method is robust to random initialization and converges faster than other clustering methods. Experiments on real image segmentation are reported. optima consist not only of the eigenvectors, but of a whole family spanned by the eigenvectors through orthonormal transforms. The goal is to find the right orthonormal transform that leads to a discretization. ˜X normalize
Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations
 Concurrency
, 1991
"... If a finite element mesh has a sufficiently regular structure, it is easy to decide in advance how to distribute the mesh among the processors of a distributedmemory parallel processor, but if the mesh is unstructured, the problem becomes much more difficult. The distribution should be made so that ..."
Abstract

Cited by 158 (3 self)
 Add to MetaCart
If a finite element mesh has a sufficiently regular structure, it is easy to decide in advance how to distribute the mesh among the processors of a distributedmemory parallel processor, but if the mesh is unstructured, the problem becomes much more difficult. The distribution should be made so that each processor has approximately equal work to do, and such that communication overhead is minimized. If the mesh is solutionadaptive, i.e. the mesh and hence the load balancing problem change discretely during execution of the code, then it is most efficient to decide the optimal mesh distribution in parallel. In this paper three parallel algorithms, Orthogonal Recursive Bisection (ORB), Eigenvector Recursive Bisection (ERB) and a simple parallelization of Simulated Annealing (SA) have been implemented for load balancing a dynamic unstructured triangular mesh on 16 processors of an NCUBE machine. The test problem is a solutionadaptive Laplace solver, with an initial mesh of 280 elements,...
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 ..."
Abstract

Cited by 144 (8 self)
 Add to MetaCart
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...
METIS  Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 2.0
, 1995
"... 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 ..."
Abstract

Cited by 122 (5 self)
 Add to MetaCart
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 standalone 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.
Spectral Partitioning: The More Eigenvectors, the Better
 PROC. ACM/IEEE DESIGN AUTOMATION CONF
, 1995
"... The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which ..."
Abstract

Cited by 69 (3 self)
 Add to MetaCart
The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widelyused spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a kway partitioning). Our result motivates a simple ordering heuristic that is a multipleeigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multiway VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.
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 ..."
Abstract

Cited by 43 (3 self)
 Add to MetaCart
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.
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
Abstract

Cited by 42 (0 self)
 Add to MetaCart
In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.