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HYPERGRAPH PARTITIONINGBASED FILLREDUCING ORDERING
, 2009
"... A typical first step of a direct solver for linear system Mx = b is reordering of symmetric matrix M to improve execution time and space requirements of the solution process. In this work, we propose a novel nesteddissectionbased ordering approach that utilizes hypergraph partitioning. Our approac ..."
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Cited by 13 (6 self)
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A typical first step of a direct solver for linear system Mx = b is reordering of symmetric matrix M to improve execution time and space requirements of the solution process. In this work, we propose a novel nesteddissectionbased ordering approach that utilizes hypergraph partitioning. Our approach is based on formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem. This new formulation is immune to deficiency of GPVS in a multilevel framework hence enables better orderings. In matrix terms, our method relies on the existence of a structural factorization of the input M matrix in the form of M = AAT (or M = AD2AT). We show that the partitioning of the rownet hypergraph representation of rectangular matrix A induces a GPVS of the standard graph representation of matrix M. In the absence of such factorization, we also propose simple, yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix M, and hence applicable to any arbitrary symmetric matrix M. Our experimental evaluation has shown that the proposed method achieves better ordering in comparison to stateoftheart graphbased ordering tools even for symmetric matrices where structural M = AAT factorization is not provided as an input. For matrices coming from linear programming problems, our method enables even faster and better orderings.
Constructing Elimination Trees for Sparse Unsymmetric Matrices
"... The elimination tree model for sparse unsymmetric matrices and an algorithm for constructing it have been recently proposed [Eisenstat and Liu, SIAM J. Matrix Anal. Appl., 26 (2005) and 29 (2008)]. The construction algorithm has a worst case time complexity O(mn) for an n × n unsymmetric matrix havi ..."
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Cited by 2 (0 self)
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The elimination tree model for sparse unsymmetric matrices and an algorithm for constructing it have been recently proposed [Eisenstat and Liu, SIAM J. Matrix Anal. Appl., 26 (2005) and 29 (2008)]. The construction algorithm has a worst case time complexity O(mn) for an n × n unsymmetric matrix having m nonzeros. We propose another algorithm that has a worst case time complexity of O(m log n). 1
Computing symmetric nonnegative rank factorizations
 Linear Algebra Appl
"... An algorithm is described for the nonnegative rank factorization (NRF) of some completely positive (CP) matrices whose rank is equal to their CPrank. The algorithm can compute the symmetric NRF of any nonnegative symmetric rankr matrix that contains a diagonal principal submatrix of that rank and ..."
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An algorithm is described for the nonnegative rank factorization (NRF) of some completely positive (CP) matrices whose rank is equal to their CPrank. The algorithm can compute the symmetric NRF of any nonnegative symmetric rankr matrix that contains a diagonal principal submatrix of that rank and size with leading cost O(rm2) operations in the dense case. The algorithm is based on geometric considerations and is easy to implement. The implications for matrix graphs are also discussed.
Searching for better fillin
, 2013
"... Minimum Fillin is a fundamental and classical problem arising in sparse matrix computations. In terms of graphs it can be formulated as a problem of finding a triangulation of a given graph with the minimum number of edges. By the classical result of Rose, Tarjan, Lueker, and Ohtsuki from 1976, an ..."
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Minimum Fillin is a fundamental and classical problem arising in sparse matrix computations. In terms of graphs it can be formulated as a problem of finding a triangulation of a given graph with the minimum number of edges. By the classical result of Rose, Tarjan, Lueker, and Ohtsuki from 1976, an inclusion minimal triangulation of a graph can be found in polynomial time but, as it was shown by Yannakakis in 1981, finding a triangulation with the minimum number of edges is NPhard. In this paper, we study the parameterized complexity of local search for the Minimum Fillin problem in the following form: Given a triangulation H of a graph G, is there a better triangulation, i.e. triangulation with less edges than H, within a given distance from H? We prove that this problem is fixedparameter tractable (FPT) being parameterized by the distance from the initial triangulation by providing an algorithm that in time O(f(k)G  O(1) ) decides if a better triangulation of G can be obtained by swapping at most k edges of H. Our result adds Minimum Fillin to the list of very few problems for which local search is known to be FPT.
TITLE ILUPACK BYLINE
"... ILUPACK is the abbreviation for Incomplete LU factorization PACKage. It is a software library for the iterative solution of large sparse linear systems. It is written in FORTRAN 77 and C and available at ..."
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ILUPACK is the abbreviation for Incomplete LU factorization PACKage. It is a software library for the iterative solution of large sparse linear systems. It is written in FORTRAN 77 and C and available at
Computing
"... 5.1. Unified model for assessing checkpointing protocols at extremescale 3 5.2. Impact of fault prediction on checkpointing strategies 3 5.3. Combining process replication and checkpointing for resilience on exascale systems 3 5.4. On the complexity of scheduling checkpoints for computational workf ..."
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5.1. Unified model for assessing checkpointing protocols at extremescale 3 5.2. Impact of fault prediction on checkpointing strategies 3 5.3. Combining process replication and checkpointing for resilience on exascale systems 3 5.4. On the complexity of scheduling checkpoints for computational workflows 4 5.5. Scheduling treeshaped task graphs to minimize memory and makespan 4 5.6. Memory allocation for different classes of DAGs 4 5.7. Scheduling nonlinear divisible loads 4 5.8. Energyaware scheduling under reliability and makespan constraints 5 5.9. Approximation algorithms for energy, reliability and makespan optimization problems 5 5.10. Optimal algorithms and approximation algorithms for replica placement with distance constraints in tree networks 5 5.11. Throughput optimization for pipeline workflow scheduling with setup times 6 5.12. Semimatching algorithms for scheduling parallel tasks under resource constraints 6 5.13. A Symmetry preserving algorithm for matrix scaling 6 5.14. On sharedmemory parallelization of a sparse matrix scaling algorithm 6
An Approximate Minimal Elimination Ordering Scheme
"... factorization Abstract. Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. In view of some known minimal elimination ordering methods, an efficient heuristic approximate minimal elimination ordering scheme is pr ..."
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factorization Abstract. Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. In view of some known minimal elimination ordering methods, an efficient heuristic approximate minimal elimination ordering scheme is proposed, which has the total running time of O(n+m). It is noteworthy that the algorithm can not only find a good ordering efficiently, but also achieve the result of symbolic factorization simultaneously. 1.