Abstract not found

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.

Unstructured meshes are used in many large-scale scientific and engineering problems, including finite-volume methods for computational fluid dynamics and finite-element methods for structural analysis. If unstructured problems such as these are to be solved on distributed-memory parallel computers, their data structures must be partitioned and distributed across processors; if they are to be solved efficiently, the partitioning must maximize load balance and minimize interprocessor communication. Recently the recursive spectral bisection method (RSB) has been shown to be very effective for such partitioning problems compared to alternative methods. Unfortunately, RSB in its simplest form is rather expensive. In this report we shall describe a multilevel implementation of RSB that can attain about an order-of-magnitude improvement in run time on typical examples. Keywords: graph partitioning, domain decomposition, MIMD machines, multilevel algorithm, spectral bisection, sp...

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In this paper, we present a new hypergraphpartitioning algorithm that is based on the multilevel paradigm. In the multilevel paradigm, a sequence of successively coarser hypergraphs is constructed. A bisection of the smallest hypergraph is computed and it is used to obtain a bisection of the original hypergraph by successively projecting and refining the bisection to the next level finer hypergraph. We have developed new hypergraph coarsening strategies within the multilevel framework. We evaluate their performance both in terms of the size of the hyperedge cut on the bisection, as well as on the run time for a number of very large scale integration circuits. Our experiments show that our multilevel hypergraph-partitioning algorithm produces high-quality partitioning in a relatively small amount of time. The quality of the partitionings produced by our scheme are on the average 6%–23 % better than those produced by other state-of-the-art schemes. Furthermore, our partitioning algorithm is significantly faster, often requiring 4–10 times less time than that required by the other schemes. Our multilevel hypergraph-partitioning algorithm scales very well for large hypergraphs. Hypergraphs with over 100 000 vertices can be bisected in a few minutes on today’s workstations. Also, on the large hypergraphs, our scheme outperforms other schemes (in hyperedge cut) quite consistently with larger margins (9%–30%).

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Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacian-based methods in a statistical setting.

We introduce the Split-C language, a parallel extension of C intended for high performance programming on distributed memory multiprocessors, and demonstrate the use of the language in optimizing parallel programs. Split-C provides a global address space with a clear concept of locality and unusual assignment operators. These are used as tools to reduce the frequency and cost of remote access. The language allows a mixture of shared memory, message passing, and data parallel programming styles while providing efficient access to the underlying machine. We demonstrate the basic language concepts using regular and irregular parallel programs and give performance results for various stages of program optimization. 1 Overview Split-C is a parallel extension of the C programming language that supports efficient access to a global address space on current distributed memory multiprocessors. It retains the "small language" character of C and supports careful engineering and optimization of ...

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,...

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