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29
Parallel Multilevel Algorithms for Hypergraph Partitioning
, 2007
"... In this paper, we present parallel multilevel algorithms for the hypergraph partitioning problem. In particular, we describe schemes for parallel coarsening, parallel greedy kway refinement and parallel multiphase refinement. Using an asymptotic theoretical performance model, we derive the isoeffi ..."
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In this paper, we present parallel multilevel algorithms for the hypergraph partitioning problem. In particular, we describe schemes for parallel coarsening, parallel greedy kway refinement and parallel multiphase refinement. Using an asymptotic theoretical performance model, we derive the isoefficiency function for our algorithms and hence show that they are technically scalable when the maximum vertex and hyperedge degrees are small. We conduct experiments on hypergraphs from six different application domains to investigate the empirical scalability of our algorithms both in terms of runtime and partition quality. Our findings confirm that the quality of partition produced by our algorithms is stable as the number of processors is increased while being competitive with those produced by a stateoftheart serial multilevel partitioning tool. We also validate our theoretical performance model through an isoefficiency study. Finally, we evaluate the impact of introducing parallel multiphase refinement into our parallel multilevel algorithm in terms of the trade off between improved partition quality and higher runtime cost.
A Parallel Matrix Scaling Algorithm
"... We recently proposed an iterative procedure which asymptotically scales the rows and columns of a given matrix to one in a given norm. In this work, we briefly mention some of the properties of that algorithm and discuss its efficient parallelization. We report on a parallel performance study of ou ..."
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We recently proposed an iterative procedure which asymptotically scales the rows and columns of a given matrix to one in a given norm. In this work, we briefly mention some of the properties of that algorithm and discuss its efficient parallelization. We report on a parallel performance study of our implementation on a few computing environments.
A General Graph Model For Representing Exact Communication Volume in Parallel Sparse Matrix–Vector Multiplication
, 2006
"... In this paper, we present a new graph model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Our model differs from previous graphbased approaches in two main respects. Firstly, our model is based on edge colouring rather than vertex partitioning. Secondly, our mod ..."
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In this paper, we present a new graph model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Our model differs from previous graphbased approaches in two main respects. Firstly, our model is based on edge colouring rather than vertex partitioning. Secondly, our model is able to correctly quantify and minimise the total communication volume of the parallel sparse matrix– vector multiplication while maintaining the computational load balance across the processors. We show that our graph edge colouring model is equivalent to the finegrained hypergraph partitioningbased sparse matrix decomposition model. We conjecture that the existence of such a graph model should lead to faster serial and parallel sparse matrix decomposition heuristics and associated tools.
WebSiteBased Partitioning Techniques for Reducing the Preprocessing Overhead before the Parallel PageRank Computations
"... Abstract. The efficiency of the PageRank computation is important since the constantly evolving nature of the Web requires this computation to be repeated many times. Due to the enormous size of the Web’s hyperlink structure, PageRank computations are usually carried out on parallel computers. Recen ..."
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Abstract. The efficiency of the PageRank computation is important since the constantly evolving nature of the Web requires this computation to be repeated many times. Due to the enormous size of the Web’s hyperlink structure, PageRank computations are usually carried out on parallel computers. Recently, a hypergraphpartitioningbased formulation for parallel sparsematrix vector multiplication is proposed as a preprocessing step which will minimize the communication overhead of the parallel PageRank computations. Based on this work, we propose Websitebased partitioning approaches in order to reduce the overhead of this preprocessing step. The conducted experiments show that the proposed approach produces comparable performance results for PageRank computation while achieving lower preprocessing overheads. 1
SiteBased Partitioning and Repartitioning Techniques for Parallel PageRank Computation
"... Abstract—The PageRank algorithm is an important component in effective web search. At the core of this algorithm are repeated sparse matrixvector multiplications where the involved web matrices grow in parallel with the growth of the web and are stored in a distributed manner due to space limitatio ..."
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Abstract—The PageRank algorithm is an important component in effective web search. At the core of this algorithm are repeated sparse matrixvector multiplications where the involved web matrices grow in parallel with the growth of the web and are stored in a distributed manner due to space limitations. Hence, the PageRank computation, which is frequently repeated, must be performed in parallel with highefficiency and lowpreprocessing overhead while considering the initial distributed nature of the web matrices. Our contributions in this work are twofold. We first investigate the application of stateoftheart sparse matrix partitioning models in order to attain high efficiency in parallel PageRank computations with a particular focus on reducing the preprocessing overhead they introduce. For this purpose, we evaluate two different compression schemes on the web matrix using the site information inherently available in links. Second, we consider the more realistic scenario of starting with an initially distributed data and extend our algorithms to cover the repartitioning of such data for efficient PageRank computation. We report performance results using our parallelization of a stateoftheart PageRank algorithm on two different PC clusters with 40 and 64 processors. Experiments show that the proposed techniques achieve considerably high speedups while incurring a preprocessing overhead of several iterations (for some instances even less than a single iteration) of the underlying sequential PageRank algorithm. Index Terms—PageRank, sparse matrixvector multiplication, web search, parallelization, sparse matrix partitioning, graph partitioning, hypergraph partitioning, repartitioning. Ç
UMPa: A Multiobjective, multilevel partitioner for communication minimization
"... Abstract. We propose a directed hypergraph model and a refinement heuristic to distribute communicating tasks among the processing units in a distributed memory setting. The aim is to achieve load balance and minimize the maximum data sent by a processing unit. We also take two other communication m ..."
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Abstract. We propose a directed hypergraph model and a refinement heuristic to distribute communicating tasks among the processing units in a distributed memory setting. The aim is to achieve load balance and minimize the maximum data sent by a processing unit. We also take two other communication metrics into account with a tiebreaking scheme. With this approach, task distributions causing an excessive use of network or a bottleneck processor which participates to almost all of the communication are avoided. We show on a large number of problem instances that our model improves the maximum data sent by a processor up to 34 % for parallel environments with 4, 16, 64 and 256 processing units compared to the state of the art which only minimizes the total communication volume.
Hypergraph Partitioning for Parallel Iterative Solution of General Sparse Linear Systems ∗
, 2007
"... The efficiency of parallel iterative methods for solving linear systems, arising from reallife applications, depends greatly on matrix characteristics and on the amount of parallel overhead. It is often viewed that a major part of this overhead can be caused by parallel matrixvector multiplications ..."
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The efficiency of parallel iterative methods for solving linear systems, arising from reallife applications, depends greatly on matrix characteristics and on the amount of parallel overhead. It is often viewed that a major part of this overhead can be caused by parallel matrixvector multiplications. However, for difficult large linear systems, the preconditioning operations needed to accelerate convergence are to be performed in parallel and may also incur substantial overhead. To obtain an efficient preconditioning, it is desirable to consider certain matrix numerical properties in the matrix partitioning process. In general, graph partitioners consider the nonzero structure of a matrix to balance the number of unknowns and to decrease communication volume among parts. The present work builds upon hypergraph partitioning techniques because of their ability to handle nonsymmetric and irregular structured matrices and because they correctly minimize communication volume. First, several hyperedge weight schemes are proposed to account for the numerical matrix property called diagonal dominance of rows and columns. Then, an algorithm for the independent partitioning of certain submatrices followed by the matching of the obtained parts is presented in detail along with a proof that it correctly minimizes the total communication volume. For the proposed variants of hypergraph partitioning models, numerical experiments compare the iterations to converge, investigate the diagonal dominance of the obtained parts, and show the values of the partitioning cost functions. 1
A matrix partitioning interface to PaToH in MATLAB
, 2009
"... We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraphbased sparse matrix partitioning methods which are used for efficient parallelization of sparse matrixvector multiplication operations. The interface also offers tools for visualizing and measur ..."
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We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraphbased sparse matrix partitioning methods which are used for efficient parallelization of sparse matrixvector multiplication operations. The interface also offers tools for visualizing and measuring the quality of a given matrix partition. We propose a novel, multilevel, 2D coarseningbased 2D matrix partitioning method and implement it using the interface. We have performed extensive comparison of the proposed method against our implementation of orthogonal recursive bisection and finegrain methods on a large set of publicly available test matrices. The conclusion of the experiments is that the new method can compete with the finegrain method while also suggesting new research directions.