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70
Multilevel algorithms for partitioning powerlaw graphs
 IEEE INTERNATIONAL PARALLEL & DISTRIBUTED PROCESSING SYMPOSIUM (IPDPS). IN
, 2006
"... Graph partitioning is an enabling technology for parallel processing as it allows for the effective decomposition of unstructured computations whose data dependencies correspond to a large sparse and irregular graph. Even though the problem of computing highquality partitionings of graphs arising i ..."
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Cited by 61 (1 self)
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Graph partitioning is an enabling technology for parallel processing as it allows for the effective decomposition of unstructured computations whose data dependencies correspond to a large sparse and irregular graph. Even though the problem of computing highquality partitionings of graphs arising in scientific computations is to a large extent wellunderstood, this is far from being true for emerging HPC applications whose underlying computation involves graphs whose degree distribution follows a powerlaw curve. This paper presents new multilevel graph partitioning algorithms that are specifically designed for partitioning such graphs. It presents new clusteringbased coarsening schemes that identify and collapse together groups of vertices that are highly connected. An experimental evaluation of these schemes on 10 different graphs show that the proposed algorithms consistently and significantly
Permuting Sparse Rectangular Matrices into BlockDiagonal Form
 SIAM Journal on Scientific Computing
, 2002
"... We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. W ..."
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Cited by 56 (18 self)
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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using stateoftheart graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
Hypergraphbased Dynamic Load Balancing for Adaptive Scientific Computations
, 2007
"... Adaptive scientific computations require that periodic repartitioning (load balancing) occur dynamically to maintain load balance. Hypergraph partitioning is a successful model for minimizing communication volume in scientific computations, and partitioning software for the static case is widely ava ..."
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Cited by 39 (6 self)
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Adaptive scientific computations require that periodic repartitioning (load balancing) occur dynamically to maintain load balance. Hypergraph partitioning is a successful model for minimizing communication volume in scientific computations, and partitioning software for the static case is widely available. In this paper, we present a new hypergraph model for the dynamic case, where we minimize the sum of communication in the application plus the migration cost to move data, thereby reducing total execution time. The new model can be solved using hypergraph partitioning with fixed vertices. We describe an implementation of a parallel multilevel repartitioning algorithm within the Zoltan loadbalancing toolkit, which to our knowledge is the first code for dynamic load balancing based on hypergraph partitioning. Finally, we present experimental results that demonstrate the effectiveness of our approach on a Linux cluster with up to 64 processors. Our new algorithm compares favorably to the widely used ParMETIS partitioning software in terms of quality, and would have reduced total execution time in most of our test cases.
Encapsulating Multiple CommunicationCost Metrics in Partitioning Sparse Rectangular Matrices for Parallel MatrixVector Multiplies
"... This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational load ..."
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Cited by 37 (22 self)
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This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational loads of processors. Most of the existing partitioning models consider only the total message volume hoping that minimizing this communicationcost metric is likely to reduce other metrics. However, the total message latency (startup time) may be more important than the total message volume. Furthermore, the maximum message volume and latency handled by a single processor are also important metrics. We propose a twophase approach that encapsulates all these four communicationcost metrics. The objective in the first phase is to minimize the total message volume while maintainingthe computationalload balance. The objective in the second phase is to encapsulate the remaining three communicationcost metrics. We propose communicationhypergraph and partitioning models for the second phase. We then present several methods for partitioning communication hypergraphs. Experiments on a wide range of test matrices show that the proposed approach yields very effective partitioning results. A parallel implementation on a PC cluster verifies that the theoretical improvements shown by partitioning results hold in practice.
On twodimensional sparse matrix partitioning: Models, methods, and a recipe
 SIAM J. SCI. COMPUT
, 2010
"... We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces finegrain ..."
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Cited by 35 (18 self)
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We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces finegrain partitions. The other two produce coarser partitions, where one of them imposes a limit on the number of messages sent and received by a single processor, and the other trades that limit for a lower communication volume. We also present a thorough experimental evaluation of the proposed twodimensional partitioning methods together with the hypergraphbased onedimensional partitioning methods, using an extensive set of public domain matrices. Furthermore, for the users of these partitioning methods, we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics.
New Challenges in Dynamic Load Balancing
 APPL. NUMER. MATH
, 2004
"... Data partitioning and load balancing are important components of parallel computations. Many different partitioning strategies have been developed, with great effectiveness in parallel applications. But the loadbalancing problem is not yet solved completely; new applications and architectures requi ..."
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Cited by 33 (6 self)
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Data partitioning and load balancing are important components of parallel computations. Many different partitioning strategies have been developed, with great effectiveness in parallel applications. But the loadbalancing problem is not yet solved completely; new applications and architectures require new partitioning features. Existing algorithms must be enhanced to support more complex applications. New models are needed for nonsquare, nonsymmetric, and highly connected systems arising from applications in biology, circuits, and materials simulations. Increased use of heterogeneous computing architectures requires partitioners that account for nonuniform computing, network, and memory resources. And, for greatest impact, these new capabilities must be delivered in toolkits that are robust, easytouse, and applicable to a wide range of applications. In this paper, we discuss our approaches to addressing these issues within the Zoltan Parallel Data Services toolkit.
A finegrain hypergraph model for 2D decomposition of sparse matrices
 in: Proceedings of the 15th International Parallel and Distributed Processing Symposium, 2001, p. 118. C. Aykanat
"... We propose a new hypergraph model for the decomposition of irregular computational domains. This work focuses on the decomposition of sparse matrices for parallel matrixvector multiplication. However, the proposed model can also be used to decompose computational domains of other parallel reduction ..."
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Cited by 33 (8 self)
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We propose a new hypergraph model for the decomposition of irregular computational domains. This work focuses on the decomposition of sparse matrices for parallel matrixvector multiplication. However, the proposed model can also be used to decompose computational domains of other parallel reduction problems. We propose a “finegrain” hypergraph model for twodimensional decomposition of sparse matrices. In the proposed finegrain hypergraph model, vertices represent nonzeros and hyperedges represent sparsity patterns of rows and columns of the matrix. By partitioning the finegrain hypergraph into equally weighted vertex parts (processors) so that hyperedges are split among as few processors as possible, the model correctly minimizes communication volume while maintaining computationalload balance. Experimental results on a wide range of realistic sparse matrices confirm the validity of the proposed model, by achieving up to 50 percent better decompositionsthan the existing models, in terms of totalcommunication volume. 1
A hypergraphpartitioning approach for coarsegrain decomposition
 in: Proceedings of the 2001 ACM/IEEE Conference on Supercomputing, 2001
"... We propose a new twophase method for the coarsegrain decomposition of irregular computational domains. This work focuses on the 2D partitioning of sparse matrices for parallel matrixvector multiplication. However, the proposed model can also be used to decompose computational domains of other par ..."
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Cited by 32 (15 self)
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We propose a new twophase method for the coarsegrain decomposition of irregular computational domains. This work focuses on the 2D partitioning of sparse matrices for parallel matrixvector multiplication. However, the proposed model can also be used to decompose computational domains of other parallel reduction problems. This work also introduces the use of multiconstraint hypergraph partitioning, for solving the decomposition problem. The proposed method explicitly models the minimization of communication volume while enforcing the upper bound of p + q; 2 on the maximum number of messages handled by a single processor, for a parallel system with P = p q processors. Experimental results on a wide range of realistic sparse matrices confirm the validity of the proposed methods, by achieving up to 25 percent better partitions than the standard graph model, in terms of total communication volume, and 59 percent better partitions in terms of number of messages, on the overall average. 1.