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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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 distributed-memory 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 solution-adaptive, 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 solution-adaptive Laplace solver, with an initial mesh of 280 elements,...

Spectral partitioning methods use the Fiedler vector---the eigenvector of the secondsmallest eigenvalue of the Laplacian matrix---to 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 bounded-degree 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 bounded-degree planar graphs and two-dimensional meshes and O i n 1=d j for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...

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 stand-alone 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.

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 widely-used spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2-way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a k-way partitioning). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.

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Adaptive local grid refinement/coarsening results in unequal distribution of workload among the processors of a parallel system. A novel method for balancing the load in cases of dynamically changing tetrahedral grids is developed. The approach Graduate Research Assistant, Dept. of Electrical and Computer Engineering y Assistant Professor, Member AIAA z Research Scientist, Member AIAA 1 employs local exchange of cells among processors in order to redistribute the load equally. An important part of the load balancing algorithm is the method employed by a processor to determine which cells within its subdomain are to be exchanged. Two such methods are presented and compared. The strategy for load balancing is based on the Divide-and-Conquer approach which leads to an efficient parallel algorithm. This method is implemented on a distributed-memory MIMD system. 1 Introduction Computational fluid dynamics (CFD) has advanced rapidly over the last two decades and it is recognized as a...

The complexity of circuit designs has necessitated a top-down approach to layout synthesis. A large body of work shows that a good layout hierarchy, or partitioning tree, as measured by the associated Rent parameter, will correspond to an area-efficient layout. We define the intrinsic Rent parameter of a netlist to be the minimum possible Rent parameter of any partitioning tree for the netlist. Experimental results show that spectra-based ratio cut partitioning algorithms yield partitioning trees with the lowest observed Rent parameter over all benchmarks and over all algorithms tested. For examples where the intrinsic Rent parameter is known, spectral ratio cut partitioning yields a partitioning tree with Rent parameter essentially identical to this theoretical optimum. These results have deep implications withrespect to both the choice of partitioning algorithms for top-down layout, as well as new approaches to layout area estimation. The paper concludes with directions for future research, including several promising techniques for fast estimation of the (intrinsic) Rent parameter.

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.