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58
A Multilevel Algorithm for ForceDirected GraphDrawing
, 2003
"... We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a forcedirected placement algorithm. ..."
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Cited by 89 (3 self)
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We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a forcedirected placement algorithm.
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.
Mesh Partitioning: a Multilevel Balancing and Refinement Algorithm
, 1998
"... Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. In this paper we present an enhancement o ..."
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Cited by 55 (22 self)
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Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. In this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions. We also present a formulation of the KernighanLin partition optimisation algorithm which incorporates loadbalancing. The resulting algorithm is tested against a different but related stateofthe art partitioner and shown to provide improved results. Keywords: graphpartitioning, mesh partitioning, loadbalancing, multilevel algorithms. 1 Introduction The need for mesh partitioning arises naturally in many finite element (FE) and finite volume (FV) applications. Meshes composed of elements such as triangles or tetrahedra are often better suited than regularly structured grids for representing completely general ge...
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...
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...
A Computational Study of Graph Partitioning
, 1993
"... Let G = (N; E) be an edgeweighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical s ..."
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Cited by 35 (10 self)
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Let G = (N; E) be an edgeweighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvaluebased techniques to find upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.
A fast kernelbased multilevel algorithm for graph clustering
 In Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
, 2005
"... Graph clustering (also called graph partitioning) — clustering the nodes of a graph — is an important problem in diverse data mining applications. Traditional approaches involve optimization of graph clustering objectives such as normalized cut or ratio association; spectral methods are widely used ..."
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Cited by 34 (3 self)
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Graph clustering (also called graph partitioning) — clustering the nodes of a graph — is an important problem in diverse data mining applications. Traditional approaches involve optimization of graph clustering objectives such as normalized cut or ratio association; spectral methods are widely used for these objectives, but they require eigenvector computation which can be slow. Recently, graph clustering with a general cut objective has been shown to be mathematically equivalent to an appropriate weighted kernel kmeans objective function. In this paper, we exploit this equivalence to develop a very fast multilevel algorithm for graph clustering. Multilevel approaches involve coarsening, initial partitioning and refinement phases, all of which may be specialized to different graph clustering objectives. Unlike existing multilevel clustering approaches, such as METIS, our algorithm does not constrain the cluster sizes to be nearly equal. Our approach gives a theoretical guarantee that the refinement step decreases the graph cut objective under consideration. Experiments show that we achieve better final objective function values as compared to a stateoftheart spectral clustering algorithm: on a series of benchmark test graphs with up to thirty thousand nodes and one million edges, our algorithm achieves lower normalized cut values in 67 % of our experiments and higher ratio association values in 100 % of our experiments. Furthermore, on large graphs, our algorithm is significantly faster than spectral methods. Finally, our algorithm requires far less memory than spectral methods; we cluster a 1.2 million node movie network into 5000 clusters, which due to memory requirements cannot be done directly with spectral methods.
A Combined Evolutionary Search and Multilevel Approach to Graph Partitioning
 London SE10 9LS
, 2000
"... Graph partitioning divides a graph into several pieces by cutting edges. The graph partitioning problem is to divide so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges separating these pieces is minimised. Important examples of th ..."
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Cited by 32 (3 self)
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Graph partitioning divides a graph into several pieces by cutting edges. The graph partitioning problem is to divide so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges separating these pieces is minimised. Important examples of the problem arise in partitioning graphs known as meshes for the parallel execution of computational mechanics codes.
Multilevel Refinement for Combinatorial Optimisation Problems
 SE10 9LS
, 2001
"... Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (some ..."
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Cited by 30 (5 self)
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Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.
A Probabilistic Framework for Relational Clustering
 KDD'07
"... Relational clustering has attracted more and more attention due to its phenomenal impact in various important applications which involve multitype interrelated data objects, such as Web mining, search marketing, bioinformatics, citation analysis, and epidemiology. In this paper, we propose a probab ..."
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Cited by 29 (1 self)
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Relational clustering has attracted more and more attention due to its phenomenal impact in various important applications which involve multitype interrelated data objects, such as Web mining, search marketing, bioinformatics, citation analysis, and epidemiology. In this paper, we propose a probabilistic model for relational clustering, which also provides a principal framework to unify various important clustering tasks including traditional attributesbased clustering, semisupervised clustering, coclustering and graph clustering. The proposed model seeks to identify cluster structures for each type of data objects and interaction patterns between different types of objects. Under this model, we propose parametric hard and soft relational clustering algorithms under a large number of exponential family distributions. The algorithms are applicable to relational data of various structures and at the same time unifies a number of statoftheart clustering algorithms: coclustering algorithms, the kpartite graph clustering, and semisupervised clustering based on hidden Markov random fields.