Permuting Sparse Rectangular Matrices into Block-Diagonal Form

  title={Permuting Sparse Rectangular Matrices into Block-Diagonal Form},
  author={Cevdet Aykanat and Ali Pinar and {\"U}mit V. Çataly{\"u}rek},
  journal={SIAM J. Sci. Comput.},
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. To represent the nonzero structure of a matrix, we propose bipartite graph and hypergraph models that reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph… 

A Recursive Bipartitioning Algorithm for Permuting Sparse Square Matrices into Block Diagonal Form with Overlap

A left-to-right bipartitioning approach together with a novel vertex fixation scheme is proposed so that existing 2-way GPVS tools that support fixed vertices can be effectively and efficiently utilized in the oGPVS problem.

Reordering sparse matrices into block-diagonal column-overlapped form

A Geometric Approach to Matrix Ordering

A recursive way to partition hypergraphs is presented which creates and exploits hypergraph geometry and is suitable for many-core parallel architectures and is on average 21.6 times faster than Mondriaan.

Fill-in reduction in sparse matrix factorizations using hypergraphs

The use of hypergraph partitioning based methods in fill-reducing orderings of sparse matrices for Cholesky, LU and QR factorizations is discussed and a recently proposed technique to use hypergraph models in a fairly standard manner is adopted.

The Effect of Various Sparsity Structures on Parallelism and Algorithms to Reveal Those Structures

This chapter presents a number of models and their relationship with parallel numerical methods, focusing especially on graph and hypergraph partitioning models in obtaining several different sparse matrix forms.

Singly-Bordered Block-Diagonal Form for Minimal Problem Solvers

The Grobner basis method for solving systems of polynomial equations became very popular in the computer vision community as it helps to find fast and numerically stable solutions to difficult problems and is potentially significantly speeds up Solvers of several important minimal computer vision problems.

Hypergraph Partitioning for Parallel Iterative Solution of General Sparse Linear Systems ∗

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.

Hypergraph Partitioning-Based Fill-Reducing Ordering for Symmetric Matrices

A novel nested-dissection-based ordering approach based on the formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem that enables better orderings of matrices coming from linear programming problems.

Hypergraph-Based Unsymmetric Nested Dissection Ordering for Sparse LU Factorization

A hypergraph-based unsymmetric nested dissection (HUND) ordering for reducing the fill-in incurred during Gaussian elimination is discussed, which 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, in particular on matrices with highly unsyMMetric structures.

TR-2006-010 Revisiting hypergraph models for sparse matrix decomposition

An elementary hypergraph model in which vertices represent the data elements of a matrix-vector multiply operation and nets encode data dependencies is set forth and a recently proposed hypergraph transformation operation is applied to devise models for 1D sparse matrix decomposition.



Computing the block triangular form of a sparse matrix

This work considers the problem of permuting the rows and columns of a rectangular or square, unsymmetric sparse matrix to compute its block triangular form, a canonical decomposition of bipartite graphs induced by a maximum matching that was discovered by Dulmage and Mendelsohn.

Decomposing Irregularly Sparse Matrices for Parallel Matrix-Vector Multiplication

Experimental results on sparse matrices confirm both the validity of the proposed hypergraph models and appropriateness of the multilevel approach to hypergraph partitioning.

Partitioning Rectangular and Structurally Unsymmetric Sparse Matrices for Parallel Processing

This paper shows that the efficient parallelization of matrix-transpose-vector product operations can be expressed in terms of partitioning bipartite graphs and introduces several algorithms for this partitioning problem and compares their performance on a set of test matrices.

Partitioning Sparse Rectangular Matrices for Parallel Processing ?

The rectangular matrix partitioning problem is formalized and several methods for solving it are discussed, including the spectral partitioning method for symmetric matrices to the rectangular case and this method is compared to three new methods | the alternating partitions method and two hybrid methods.

Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication

In this work, we show that the standard graph-partitioning-based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrix-vector

Ordering Unsymmetric Matrices into Bordered Block Diagonal Form for Parallel Processing

A multilevel ordering algorithm for reordering the unsymmetric matrix into a bordered block-diagonal (BBD) form to speed up the solution of linear systems of linear equations.

Decomposing Linear Programs for Parallel Solution

This paper considers decomposing LP constraint matrices to obtain block angular structures with specified number of blocks for scalable parallelization and proposes hypergraph models to represent LP constraint Matrices for decomposition.

A Heuristic for Reducing Fill-In in Sparse Matrix Factorization

We present a heuristic that helps to improve the quality of the bisection returned by the Kernighan-Lin and greedy graph bisection algorithms. This in turns helps to reduce the amount of fill-in


It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph, which can be used to compute good separators in grid graphs.

A multilevel algorithm for partitioning graphs

A multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs, and the smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs.