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HYPERGRAPHBASED UNSYMMETRIC NESTED DISSECTION ORDERING FOR SPARSE LU FACTORIZATION
"... Abstract. In this paper we present HUND, a hypergraphbased unsymmetric nested dissection ordering algorithm for reducing the fillin incurred during Gaussian elimination. HUND has several important properties. It takes a global perspective of the entire matrix, as opposed to local heuristics. It ta ..."
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Abstract. In this paper we present HUND, a hypergraphbased unsymmetric nested dissection ordering algorithm for reducing the fillin incurred during Gaussian elimination. HUND has several important properties. It takes a global perspective of the entire matrix, as opposed to local heuristics. It takes into account the assymetry of the input matrix by using a hypergraph to represent its structure. It is suitable for performing Gaussian elimination in parallel, with partial pivoting. This is possible because the row permutations performed due to partial pivoting do not destroy the column separators identified by the nested dissection approach. Experimental results on 27 medium and large size highly unsymmetric matrices compare HUND to four other wellknown reordering algorithms. The results show that HUND provides a robust reordering algorithm, in the sense that it is the best or close to the best (often within 10%) of all the other methods.
Multithreaded Clustering for Multilevel Hypergraph Partitioning
 in IPDPS
, 2012
"... Abstract—Requirements for efficient parallelization of many complex and irregular applications can be cast as a hypergraph partitioning problem. The currentstateofthe art software libraries that provide tool support for the hypergraph partitioning problem are designed and implemented before the ..."
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Abstract—Requirements for efficient parallelization of many complex and irregular applications can be cast as a hypergraph partitioning problem. The currentstateofthe art software libraries that provide tool support for the hypergraph partitioning problem are designed and implemented before the gamechanging advancements in multicore computing. Hence, analyzing the structure of those tools for designing multithreaded versions of the algorithms is a crucial tasks. The most successful partitioning tools are based on the multilevel approach. In this approach, a given hypergraph is coarsened to a much smaller one, a partition is obtained on the the smallest hypergraph, and that partition is projected to the original hypergraph while refining it on the intermediate hypergraphs. The coarsening operation corresponds to clustering the vertices of a hypergraph and is the most time consuming task in a multilevel partitioning tool. We present three efficient multithreaded clustering algorithms which are very suited for multilevel partitioners. We compare their performance with that of the ones currently used in today’s hypergraph partitioners. We show on a large number of real life hypergraphs that our implementations, integrated into a commonly used partitioning library PaToH, achieve good speedups without reducing the clustering quality. KeywordsMultilevel hypergraph partitioning; coarsening; multithreaded clustering algorithms; multicore programming I.
Partitioning, ordering, and load balancing in a hierarchically parallel hybrid linear solver
 International Journal of High Performance Computing Applications
, 2012
"... PDSLin is a generalpurpose algebraic parallel hybrid (direct/iterative) linear solver based on the Schur complement method. The most challenging step of the solver is the computation of a preconditioner based on an approximate global Schur complement. We investigate two combinatorial problems to en ..."
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PDSLin is a generalpurpose algebraic parallel hybrid (direct/iterative) linear solver based on the Schur complement method. The most challenging step of the solver is the computation of a preconditioner based on an approximate global Schur complement. We investigate two combinatorial problems to enhance PDSLin’s performance at this step. The first is a multiconstraint partitioning problem to balance the workload while computing the preconditioner in parallel. For this, we describe and evaluate a number of graph and hypergraph partitioning algorithms to satisfy our particular objective and constraints. The second problem is to reorder the sparse righthand side vectors to improve the data access locality during the parallel solution of a sparse triangular system with multiple righthand sides. This is to speed up the process of eliminating the unknowns associated with the interface. We study two reordering techniques: one based on a postordering of the elimination tree and the other based on a hypergraph partitioning. To demonstrate the effect of these techniques on the performance of PDSLin, we present the numerical results of solving largescale linear systems arising from two applications of our interest: numerical simulations of modeling accelerator cavities and of modeling fusion devices. 1
A GEOMETRIC APPROACH TO MATRIX ORDERING
"... Abstract. We present a recursive way to partition hypergraphs which creates and exploits hypergraph geometry and is suitable for manycore parallel architectures. Such partitionings are then used to bring sparse matrices in a recursive Bordered Block Diagonal form (for processoroblivious parallel L ..."
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Abstract. We present a recursive way to partition hypergraphs which creates and exploits hypergraph geometry and is suitable for manycore parallel architectures. Such partitionings are then used to bring sparse matrices in a recursive Bordered Block Diagonal form (for processoroblivious parallel LU decomposition) or recursive Separated Block Diagonal form (for cacheoblivious sparse matrix–vector multiplication). We show that the quality of the obtained partitionings and orderings is competitive by comparing obtained fillin for LU decomposition with SuperLU (with better results for 8 of the 28 test matrices) and comparing cut sizes for sparse matrix–vector multiplication with Mondriaan (with better results for 4 of the 12 test matrices). The main advantage of the new method is its speed: it is on average 21.6 times faster than Mondriaan.
Advances in Parallel Partitioning, Load Balancing and Matrix Ordering for Scientific Computing
"... Abstract. We summarize recent advances in partitioning, load balancing, and matrix ordering for scientific computing performed by members of the CSCAPES SciDAC institute. 1. ..."
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Abstract. We summarize recent advances in partitioning, load balancing, and matrix ordering for scientific computing performed by members of the CSCAPES SciDAC institute. 1.
A RECURSIVE BIPARTITIONING ALGORITHM FOR PERMUTING SPARSE SQUARE MATRICES INTO BLOCK DIAGONAL FORM WITH OVERLAP
, 2013
"... We investigate the problem of symmetrically permuting a square sparse matrix into a block diagonal form with overlap. This permutation problem arises in the parallelization of an explicit formulation of the multiplicative Schwarz preconditioner and a more recent block overlapping banded linear solv ..."
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We investigate the problem of symmetrically permuting a square sparse matrix into a block diagonal form with overlap. This permutation problem arises in the parallelization of an explicit formulation of the multiplicative Schwarz preconditioner and a more recent block overlapping banded linear solver as well as its application to general sparse linear systems. In order to formulate this permutation problem as a graph theoretical problem, we define a constrained version of the multiway graph partitioning by vertex separator (GPVS) problem, which is referred to as the ordered GPVS (oGPVS) problem. However, existing graph partitioning tools are unable to solve the oGPVS problem. So, we also show how the recursive bipartitioning framework can be utilized for solving the oGPVS problem. For this purpose, we propose a lefttoright bipartitioning approach together with a novel vertex fixation scheme so that existing 2way GPVS tools that support fixed vertices can be effectively and efficiently utilized in the recursive bipartitioning framework. Experimental results on a wide range of matrices confirm the validity of the proposed approach. Key words. sparse square matrices, block diagonal form with overlap, graph partitioning by vertex separator, recursive bipartitioning, partitioning with fixed vertices, combinatorial scientific computing
Hypergraph Sparsification and Its Application to Partitioning
"... Abstract—The data one needs to cope to solve today’s problems is large scale, so are the graphs and hypergraphs used to model it. Today, we have BigData, big graphs, big matrices, and in the future, they are expected to be bigger and more complex. Many of today’s algorithms will be, and some already ..."
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Abstract—The data one needs to cope to solve today’s problems is large scale, so are the graphs and hypergraphs used to model it. Today, we have BigData, big graphs, big matrices, and in the future, they are expected to be bigger and more complex. Many of today’s algorithms will be, and some already are, expensive to run on large datasets. In this work, we analyze a set of efficient techniques to make “big data”, which is modeled as a hypergraph, smaller so that its processing takes much less time. As an application use case, we take the hypergraph partitioning problem, which has been successfully used in many practical applications for various purposes including parallelization of complex and irregular applications, sparse matrix ordering, clustering, community detection, query optimization, and improving cache locality in sharedmemory systems. We conduct several experiments to show that our techniques greatly reduce the cost of the partitioning process and preserve the partitioning quality. Although we only measured their performance from the partitioning point of view, we believe the proposed techniques will be beneficial also for other applications using hypergraphs. KeywordsHypergraph sparsification; hypergraph partitioning; multilevel approach; identical nets; identical vertices; Jaccard similarity. I.