Abstract not found

Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity, and provide excellent graph partitions. However, there exists little theoretical analysis that could explain the ability of multilevel algorithms to produce good partitions. In this paper we present such an analysis. We show under certain reasonable assumptions that even if no refinement is used in the uncoarsening phase, a good bisection of the coarser graph is worse than a good bisection of the finer graph by at most a small factor. We also show that the size of a good vertex-separator of the coarse graph projected to the finer graph (without performing refinement in the uncoarsening phase) is higher than the size of a good vertexseparator of the finer graph by at most a small factor.

Abstract not found

Contents 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2 Modeling Mesh-based Computations as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.3 Static Graph Partitioning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3.1 Geometric Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.3.2 Combinatorial Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0.3.3 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.3.4 Multilevel Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0.3.5 Combined Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 0.3.6 Qualitative Comparison of Graph Partitioning Schemes . . . . . . . . . . . . . . . . . 16 0.4 Load Balancing of Adaptive Computations . . . . . .

this paper, we have focused on describing and comparing several load balancing schemes. Comparisons by timing are difficult, since times vary between runs having the same parameters. The high-speed switch of the IBM SP2 computer is a shared resource that affects run times. More subtle effects can result from differences in the order in which messages used for migration are processed. Changes in the order in which those messages are received and integrated into the local MDB result in different traversal orders of the mesh entities. These differences cause small changes in load balancings and coarsenings. While such differences in meshes and partitionings do not affect the solution accuracy, they can cause sufficient changes in efficiency to make precise timings difficult. Qualitatively, PSIRB produced the best partitions (measured as a function of total analysis time). Octree-generated partitions were comparable but resulted in slightly longer solution times. In both cases, one or two iterations of partition boundary smoothing led to a quality improvement. ITB by itself resulted in poorer partition quality, but is useful when mesh changes are small between computational stages. Predictive enrichment provided su21 perior performance to our current enrichment process with transient problems where there are frequent enrichment and balancing steps. Enhancements to the existing load balancing procedures and the implementation of new ones are under investigation. Improvements in the slice-by-slice technique used by ITB for migration are necessary. Experiments with geometrical methods that use the spatial location of elements relative to the centroids of sending and receiving processors showed promise at reducing the number of processor interconnections. Vidwans et al. [39] pr...

Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is a restriction of this technique in which mesh entities are only allowed to migrate between subdomains in one direction. The third treats the gain as a potential field and uses the concept of relative gain for selecting appropriate vertices to migrate. The results are compared and seen to produce very high global quality partitions, very rapidly. The results are also compared with another partitioning tool and shown to be of higher quality although taking longer to compute. 2000 Elsevier Science B.V. All rights reserved.

. In many important computational mechanics applications, the computation adapts dynamically during the simulation. Examples include adaptive mesh refinement, particle simulations and transient dynamics calculations. When running these kinds of simulations on a parallel computer, the work must be assigned to processors in a dynamic fashion to keep the computational load balanced. A number of approaches have been proposed for this dynamic load balancing problem. This paper reviews the major classes of algorithms, and discusses their relative merits on problems from computational mechanics. Shortcomings in the state-of-the-art are identified and suggestions are made for future research directions. Key words. dynamic load balancing, parallel computer, adaptive mesh refinement 1. Introduction. The efficient use of a parallel computer requires two, often competing, objectives to be achieved. First, the processors must be kept busy doing useful work. And second, the amount of interprocess...

In this paper we present a parallel formulation of a multilevel k-way graph partitioning algorithm, that is particularly suited for message-passing libraries that have high latency. The multilevel k-way partitioning algorithm reduces the size of the graph by successively collapsing vertices and edges (coarsening phase), finds a k-way partitioning of the smaller graph, and then it constructs a k-way partitioning for the original graph by projecting and refining the partition to successively finer graphs (uncoarsening phase). Our algorithm is able to achieve a high degree of concurrency, while maintaining the high quality partitions produced by the serial algorithm.

This paper describes the derivation of an empirically efficient parallel two-dimensional Delaunay triangulation program from a theoretically efficient CREW PRAM algorithm. Compared to previous work, the resulting implementation is not limited to datasets with a uniform distribution of points, achieves significantly better speedups over good serial code, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for a loosely-coupled cluster of workstations, a distributed-memory multicomputer, and a shared-memory multiprocessor. The Machiavelli toolkit used to transform the nested data parallelism inherent in the divide-and-conquer algorithm into achievable task and data parallelism is also described and compared to previous techniques.

This paper surveys graph partitioning algorithms used for parallel computing, with an emphasis on the problem of distributing workloads for parallel computations. Geometric, structural, and refinementbased algorithms are described and contrasted. In addition, multilevel partitioning techniques and issues related to parallel partitioning are addressed. All algorithms are evaluated qualitatively in terms of their execution speed and ability to generate partitions with small separators. 1 Introduction In its most general form, the graph partitioning problem asks how best to divide a graph's vertices into a specified number of subsets such that: (i) the number of vertices per subset is equal and (ii) the number of edges straddling the subsets is minimized. Graph partitioning has several important applications in Computer Science, including VLSI circuit layout [8], image processing [43], solving sparse linear systems, computing fill-reducing orderings for sparse matrices, and distribu...