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175
A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems, in
- Proc. 6th SIAM Conf. Parallel Processing for Scientific Computing,
, 1993
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Spectral partitioning works: planar graphs and finite element meshes, in:
- Proceedings of the 37th Annual Symposium on Foundations of Computer Science,
, 1996
"... Abstract Spectral partitioning methods use the Fiedler vector-the eigenvector of the second-smallest 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 wo ..."
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Cited by 201 (10 self)
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Abstract Spectral partitioning methods use the Fiedler vector-the eigenvector of the second-smallest 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( √ n) for bounded-degree planar graphs and two-dimensional meshes and O(n 1/d ) 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: we prove a bound of O(1/n) for bounded-degree planar graphs and O(1/n 2/d ) for well-shaped d-dimensional meshes.
Some Applications of Laplace Eigenvalues of Graphs
- GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 129 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Geometric Mesh Partitioning: Implementation and Experiments
"... We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain ..."
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Cited by 112 (20 self)
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We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of “well-shaped” finite element meshes have good separators. The geometric method is quite simple to implement: we describe a Matlab code for it in some detail. The method is also quite efficient and effective: we compare it with some other methods, including spectral bisection.
How Good is Recursive Bisection?
- SIAM J. Sci. Comput
, 1995
"... . The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Becaus ..."
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Cited by 100 (5 self)
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. The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NP-complete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a p-way partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as well-shaped finite element and finite difference...
Parallel Dynamic Graph Partitioning for Adaptive Unstructured Meshes
, 1997
"... this paper we describe such a parallel optimization technique ..."
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Cited by 96 (18 self)
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this paper we describe such a parallel optimization technique
A Practical Approach to Dynamic Load Balancing
- IEEE Transactions on Parallel and Distributed Systems
, 1998
"... Abstract—This paper presents a cohesive, practical load balancing framework that improves upon existing strategies. These techniques are portable to a broad range of prevalent architectures, including massively parallel machines, such as the Cray T3D/E and Intel Paragon, shared memory systems, such ..."
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Cited by 82 (8 self)
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Abstract—This paper presents a cohesive, practical load balancing framework that improves upon existing strategies. These techniques are portable to a broad range of prevalent architectures, including massively parallel machines, such as the Cray T3D/E and Intel Paragon, shared memory systems, such as the Silicon Graphics PowerChallenge, and networks of workstations. As part of the work, an adaptive heat diffusion scheme is presented, as well as a task selection mechanism that can preserve or improve communication locality. Unlike many previous efforts in this arena, the techniques have been applied to two large-scale industrial applications on a variety of multicomputers. In the process, this work exposes a serious deficiency in current load balancing strategies, motivating further work in this area. Index Terms—Dynamic load balancing, diffusion, massively parallel computing, irregular problems. ————————— — F —————————— 1
An Efficient Partitioning Algorithm for Distributed Virtual Environment Systems
- IEEE TRANS. PARALLEL AND DISTRIBUTED SYSTEMS
, 2002
"... Distributed 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 ge ..."
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Cited by 71 (0 self)
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Distributed 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.
Dynamic Load Balancing for PDE Solvers on Adaptive Unstructured Meshes
, 1992
"... Modern PDE solvers written for time-dependent problems increasingly employ adaptive unstructured meshes (see Flaherty et al. [4]) in order to both increase efficiency and control the numerical error. If a distributed memory parallel computer is to be used, there arises the significant problem of div ..."
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Cited by 65 (15 self)
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Modern PDE solvers written for time-dependent problems increasingly employ adaptive unstructured meshes (see Flaherty et al. [4]) in order to both increase efficiency and control the numerical error. If a distributed memory parallel computer is to be used, there arises the significant problem of dividing up the domain equally amongst the processors whilst minimising the intersubdomain dependencies. A number of graph based algorithms have recently been proposed for steady state calculations, for example [6] & [11]. This paper considers an extension to such methods which renders them more suitable for time-dependent problems in which the mesh may be changed frequently. 1 Introduction Modern PDE solvers for time-dependent applications are currently being written so as to obtain accurate solutions to real-life problems with the solution process as automatic as possible. The use of an unstructured mesh allows the code to cater for completely general geometries and hence a wide range of pro...
Run-time and compile-time support for adaptive irregular problems
- SUPERCOMPUTING’94
, 1994
"... In adaptive irregular problems the data arrays are accessed via indirection arrays, and data access patterns change during computation. Implementing such problems on distributed memory machines requires support for dynamic data partitioning, efficient preprocessing and fast data migration. This rese ..."
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Cited by 63 (9 self)
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In adaptive irregular problems the data arrays are accessed via indirection arrays, and data access patterns change during computation. Implementing such problems on distributed memory machines requires support for dynamic data partitioning, efficient preprocessing and fast data migration. This research presents efficient runtime primitives for such problems. This new set of primitives is part of the CHAOS library. It subsumes the previous PARTI library which targeted only static irregular problems. To demonstrate the efficacy of the runtime support, two real adaptive irregular applications have been parallelized using CHAOS primitives: a molecular dynamics code (CHARMM) and a particle-in-cell code (DSMC). The paper also proposes extensions to Fortran D which can allow compilers to generate more efficient code for adaptive problems. These language extensions have been implemented in the Syracuse Fortran 90D/HPF prototype compiler. The performance of the compiler parallelized codes is compared with the hand parallelized versions.
