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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 37 (21 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.
Multilevel direct Kway hypergraph partitioning with multiple constraints and fixed vertices
, 2007
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Revisiting hypergraph models for sparse matrix partitioning
 SIAM Review
, 2007
"... Abstract. We provide an exposition of hypergraph models for parallelizing sparse matrixvector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for the parallel matrixvector multiply based on onedimensional (1D) matri ..."
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Cited by 22 (13 self)
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Abstract. We provide an exposition of hypergraph models for parallelizing sparse matrixvector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for the parallel matrixvector multiply based on onedimensional (1D) matrix partitioning. In the elementary model, the vertices represent the data of a matrixvector multiply, and the nets encode dependencies among the data. We then apply a recently proposed hypergraph transformation operation to devise models for 1Dsparse matrix partitioning. The resulting 1Dpartitioning models are equivalent to the previously proposed computational hypergraph models and are not meant to be replacements for them. Nevertheless, the new models give us insights into the previous ones and help us explain a subtle requirement, known as the consistency condition, of hypergraph partitioning models. Later, we demonstrate the flexibility of the elementary model on a few 1Dpartitioning problems that are hard to solve using the previously proposed models. We also discuss extensions of the proposed elementary model to twodimensional matrix partitioning. Key words. parallel computing, sparse matrixvector multiply, hypergraph models
Combinatorial problems in solving linear systems
, 2009
"... Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. ..."
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Cited by 10 (4 self)
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Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today’s numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.
Mapping TightlyCoupled Applications on Volatile Resources
"... Abstract—Platforms that comprise volatile processors, such as desktop grids, have been traditionally used for executing independenttask applications. In this work we study the scheduling of tightlycoupled iterative masterworker applications onto volatile processors. The main challenge is that work ..."
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Cited by 2 (1 self)
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Abstract—Platforms that comprise volatile processors, such as desktop grids, have been traditionally used for executing independenttask applications. In this work we study the scheduling of tightlycoupled iterative masterworker applications onto volatile processors. The main challenge is that workers must be simultaneously available for the application to make progress. We consider two additional complications: one should take into account that workers can become temporarily reclaimed and, for dataintensive applications, one should account for the limited bandwidth between the master and the workers. In this context, our first contribution is a theoretical study of the scheduling problem in its offline version, i.e., when processor availability is known in advance. Even in this case the problem is NPhard. Our second contribution is an analytical approximation of the expectation of the time needed by a set of workers to complete a set of tasks and of the probability of success of this computation. This approximation relies on a Markovian assumption for the temporal availability of processors. Our third contribution is a set of heuristics, some of which use the above approximation to favor reliable processors in a sensible manner. We evaluate these heuristics in simulation. We identify some heuristics that significantly outperform their competitors and derive heuristic design guidelines. I.
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.
Mapping Applications on Volatile Resources
, 2013
"... In this paper, we study the execution of iterative applications on volatile processors such as those found on desktop grids. We envision two models, one where all tasks are assumed to be independent, and another where all tasks are tightly coupled and keep exchanging information throughout the iter ..."
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In this paper, we study the execution of iterative applications on volatile processors such as those found on desktop grids. We envision two models, one where all tasks are assumed to be independent, and another where all tasks are tightly coupled and keep exchanging information throughout the iteration. These two models cover the two extreme points of the parallelization spectrum. We develop masterworker scheduling schemes that attempt to achieve good tradeoffs between worker speed and worker availability. Any iteration entails the execution of a fixed number of independent tasks or of tightlycoupled tasks. A key feature of our approach is that we consider a communication model where the bandwidth capacity of the master for sending application data to workers is limited. This limitation makes the scheduling problem more difficult both in a theoretical sense and in a practical sense. Furthermore, we consider that a processor can be in one of three states: available, down, or temporarily preempted by its owner. This preempted state also complicates the scheduling problem. In practical settings, e.g., desktop grids, master bandwidth is limited and processors are temporarily reclaimed. Consequently, addressing the aforementioned difficulties is necessary for successfully deploying masterworker applications on volatile platforms.
Contents lists available at ScienceDirect Information Sciences
"... journal homepage: www.elsevier.com/locate/ins Efficient successor retrieval operations for aggregate query processing ..."
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journal homepage: www.elsevier.com/locate/ins Efficient successor retrieval operations for aggregate query processing