We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm.

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 Kernighan-Lin partition optimisation algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-ofthe -art partitioner and shown to provide improved results. Keywords: graph-partitioning, mesh partitioning, load-balancing, 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...

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 fill-reducing ordering is crucial in the success of the numerical solution of sparse linear systems. For symmetric positive-definite 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...

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 state-of-the-art 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.

Let G = (N; E) be an edge-weighted 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 eigenvalue-based 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.

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...

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 k-means 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 state-of-the-art 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.

Most state-of-the-art ordering schemes for sparse matrices are a hybrid of a bottom-up method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottom-up and topdown methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottom-up ordering. As a consequence, we are using bottomup techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottom-up orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other popular ordering codes.

We consider the load-balancing problems which arise from parallel scientific codes containing multiple computational phases, or loops over subsets of the data, which are separated by global synchronisation points. We motivate, derive and describe the implementation of an approach which we refer to as the multiphase mesh partitioning strategy to address such issues. The technique is tested on several examples of meshes, both real and artificial, containing multiple computational phases and it is demonstrated that our method can achieve high quality partitions where a standard mesh partitioning approach fails.

Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem for distributing unstructured meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. To date these algorithms have been used almost exclusively to minimise the cut edge weight in the graph with the aim of minimising the parallel communication overhead, but recently there has been a perceived need to take into account the communications network of the parallel machine. For example the increasing use of SMP clusters (systems of multiprocessor compute nodes with very fast intra-node communications but relatively slow inter-node networks) suggest the use of hierarchical network models. Indeed this requirement is exacerbated in the early experiments with meta-computers (multiple supercomputers combined together, in extreme cases over inter-continental networks). In this paper therefore, we modify a multilevel algorithm in order to minimise a cost function based on a model of the communications network. Several network models and variants of the algorithm are tested and we establish that it is possible to successfully guide the optimisation to reflect the chosen architecture. 2001 Elsevier Science B.V. All rights reserved.