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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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.

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AbstractÐDistributed virtual environment (DVE) systems model and simulate the activities of thousands of entities interacting in a virtual world over a wide area network. Possible applications for DVE systems are multiplayer video games, military and industrial trainings, and collaborative engineering. In general, a DVE system is composed of many servers and each server is responsible to manage multiple clients who want to participate in the virtual world. Each server receives updates from different clients (such as the current position and orientation of each client) and then delivers this information to other clients in the virtual world. The server also needs to perform other tasks, such as object collision detection and synchronization control. A large scale DVE system needs to support many clients and this imposes a heavy requirement on networking resources and computational resources. Therefore, how to meet the growing requirement of bandwidth and computational resources is one of the major challenges in designing a scalable and cost-effective DVE system. In this paper, we propose an efficient partitioning algorithm that addresses the scalability issue of designing a large scale DVE system. The main idea is to dynamically divide the virtual world into different partitions and then efficiently assign these partitions to different servers. This way, each server will process approximately the same amount of workload. Another objective of the partitioning algorithm is to reduce the server-to-server communication overhead. The theoretical foundation of our dynamic partitioning algorithm is based on the linear optimization principle. We also illustrate how one can parallelize the proposed partitioning algorithm so that it can efficiently partition a very large scale DVE system. Lastly, experiments are carried out to illustrate the effectiveness of the proposed partitioning algorithm under various settings of the virtual world. Index TermsÐDistributed virtual environment, scalability issue, partitioning algorithm, load balancing, communication reduction, linear optimization. 1

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.

Abstract – In this paper we present experiments with 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. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a new scheme for refining during uncoarsening that is much faster than the Kernighan-Lin refinement. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme consistently produces partitions that are better than those produced by spectral partitioning schemes in substantially smaller timer (10 to 35 times faster than multilevel spectral bisection). Also, when our scheme is used to compute fill reducing orderings for sparse matrices, it substantially outperforms the widely used multiple minimum degree algorithm. 1

The problems of mapping and load balancing applications on arbitrary networks are considered. A novel diffusion algorithm is presented to solve the mapping problem. It complements the well known diffusion algorithms for load balancing which have enjoyed success on massively parallel computers (MPPs). Mapping is more difficult on interconnection networks than on MPPs because of the variations which occur in network topology. Popular mapping algorithms for MPPs which depend on recursive topologies are not applicable to irregular networks. The most celebrated of these MPP algorithms use information from the Laplacian matrix of a graph of communicating processes. The diffusion algorithm presented in this paper is also derived from this Laplacian matrix. The diffusion algorithm works on arbitrary network topologies and is dramatically faster than the celebrated MPP algorithms. It is delay and fault tolerant. Time to convergence depends on initial conditions and is insensitive to problem sca...

Existing partitioning algorithms provide limited support for load balancing simulations that are performed on heterogeneous parallel computing platforms. On such architectures, effective load balancing can only be achieved if the graph is distributed so that it properly takes into account the available resources (CPU speed, network bandwidth). With heterogeneous technologies becoming more popular, the need for suitable graph partitioning algorithms is critical. We developed such algorithms that can address the partitioning requirements of scientific computations, and can correctly model the architectural characteristics of emerging hardware platforms.

Abstract. In this paper, we propose a prefix code matching parallel load-balancing method Ž PCMPLB. to efficiently deal with the load imbalance of solution-adaptive finite element application programs on distributed memory multicomputers. The main idea of the PCMPLB method is first to construct a prefix code tree for processors. Based on the prefix code tree, a schedule for performing load transfer among processors can be determined by concurrently and recursively dividing the tree into two subtrees and finding a maximum matching for processors in the two subtrees until the leaves of the prefix code tree are reached. We have implemented the PCMPLB method on an SP2 parallel machine and compared its performance with two load-balancing methods, the directed diffusion method and the multilevel diffusion method, and five mapping methods, the AE�ORB method, the AE�MC method, the ML kP method, the PARTY library method, and the JOSTLE-MS method. An unstructured finite element graph Truss was used as a test sample. During the execution, Truss was refined five times. Three criteria, the execution time of mapping�load-balancing methods, the execution time of an application program under different mapping�load-balancing methods, and the speedups achieved by mapping� load-balancing methods for an application program, are used for the performance evaluation. The experimental results show that Ž. 1 if a mapping method is used for the initial partitioning and this