## Lecture 28: Sparse matrix computations

### BibTeX

@MISC{Johnsson_lecture28:,

author = {Professor S. Lennart Johnsson},

title = {Lecture 28: Sparse matrix computations},

year = {}

}

### OpenURL

### Abstract

A sparse matrix is a matrix with relatively few nonzero elements. Often the number of nonzeroes amounts to much less than 1%. Sparse matrices arises in describing network problems in matrix form. The matrix typically captures the relationship between variables in different nodes of the network. The network may be given, as in various network flow problems for electric utilities, circuit simulation, oil or gas pipelines, or transportation problems with a fixed set of routes, as in urban transportation systems. The network may also arise due to the discretization of a continuum, as in the solution of partial differential equations. But, sparse matrices may also arise in other than network problems. For instance, a sparse matrix may be used to model the relationship between objects of some type in statistical analysis, or in molecular structures. In network problems, the number of nonzero elements in a row or column is directly related to the number of neighboring nodes. This number is often independent of the total size of the network. For sparse problems originating from partial differential equations, larger matrices are typically due to a refined discretization, or the treatment of bigger problems, neither of which have any direct influence on the number of neighbors of a node. The dependence remains local. The shape of the domain, or the nature of the solution are often having the strongest influence

### Citations

538 |
Direct Methods for Sparse Matrices
- Du, Erisman, et al.
- 1986
(Show Context)
Citation Context ...ar graph. Though designed for symmetric graphs, minimum degree ordering can also be used for unsymmetric graphs. For an additional discussion of minimum degree orderings and their implementation, see =-=[7]-=-. The Cuthill and McKee ordering [4] and the Reverse Cuthill and McKee ordering used by George [13]. The basic idea in the Cuthill and McKee ordering is to build a level structure defined by picking a... |

388 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ... 23 24 25 ✉ ✉ ✉ ✉ ✉ Figure 13: Labeling of a triangulated graph. regular grids as well as for planar graphs, one of these techniques, known as nested dissection [12, 27] has been proven to be optimal =-=[19, 20]-=-. It has been shown that any bandwidth or profile ordering for N × N in two dimensions require O(N 3 ) storage and O(N 4 ) operations [22, 27]. In three dimensions, any band or profile ordering for a ... |

294 |
Algorithmic aspects of vertex elimination on graphs
- Rose, Tarjan, et al.
- 1976
(Show Context)
Citation Context .... Similarly, the graph in Figure 11 also had an elimination order that resulted in no fill–in. Unfortunately, finding an elimination order that minimizes fill–in for an arbitrary graph is NP–complete =-=[26, 25]-=-. The type of graph for which there exist an ordering that generates no fill–in during elimination is known as perfect elimination graphs [16, 1, 2, 5, 29, 15]. Determining whether or not a graph is a... |

213 |
Reducing the bandwidth of sparse symmetric matrices
- Cuthill, McKee
- 1969
(Show Context)
Citation Context ...ric graphs, minimum degree ordering can also be used for unsymmetric graphs. For an additional discussion of minimum degree orderings and their implementation, see [7]. The Cuthill and McKee ordering =-=[4]-=- and the Reverse Cuthill and McKee ordering used by George [13]. The basic idea in the Cuthill and McKee ordering is to build a level structure defined by picking a starting node for level one. The ne... |

182 |
Generalized nested dissection
- Lipton, Rose, et al.
- 1979
(Show Context)
Citation Context ... 23 24 25 ✉ ✉ ✉ ✉ ✉ Figure 13: Labeling of a triangulated graph. regular grids as well as for planar graphs, one of these techniques, known as nested dissection [12, 27] has been proven to be optimal =-=[19, 20]-=-. It has been shown that any bandwidth or profile ordering for N × N in two dimensions require O(N 3 ) storage and O(N 4 ) operations [22, 27]. In three dimensions, any band or profile ordering for a ... |

177 |
Nested dissection of regular finite element mesh
- George
- 1973
(Show Context)
Citation Context ... 20 ✉ ✉ ✉ ✉ ✉ � �� � �� � �� � �� 21 22 23 24 25 ✉ ✉ ✉ ✉ ✉ Figure 13: Labeling of a triangulated graph. regular grids as well as for planar graphs, one of these techniques, known as nested dissection =-=[12, 27]-=- has been proven to be optimal [19, 20]. It has been shown that any bandwidth or profile ordering for N × N in two dimensions require O(N 3 ) storage and O(N 4 ) operations [22, 27]. In three dimensio... |

102 |
A graph-theoretic study of the numerical solution of sparse positive-definite systems of linear equations
- Rose
- 1972
(Show Context)
Citation Context ...nimizes fill–in for an arbitrary graph is NP–complete [26, 25]. The type of graph for which there exist an ordering that generates no fill–in during elimination is known as perfect elimination graphs =-=[16, 1, 2, 5, 29, 15]-=-. Determining whether or not a graph is a perfect elimination graph can be made in time O(NM) [25]. The fill–in once an ordering is found can also be computed in time O(NM). But, finding the minimum f... |

86 |
Direct solutions of sparse network equations by optimally ordered triangular factorization
- Tinney, Walker
- 1967
(Show Context)
Citation Context ...most common orderings used for factorization are some variation of one of the following three methods For • Minimum degree • Nested dissection • Cuthill and McKee ordering The minimum degree ordering =-=[30]-=- was introduced for symmetric matrices and has its name from the fact that nodes are eliminated in order of increasing degree. Thus, for the star graph, the minimum degree ordering is optimum. In fact... |

75 |
The planar Hamiltonian circuit problem is NPcomplete
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ... except in the trivial case of a path for which the incidence matrix is tridiagonal. The NP–completeness of the bandwidth minimization problem was first proved by Papadimitriou [23]. Gary and Johnson =-=[11]-=- gave a linear time algorithm for finding the minimum bandwidth ordering of a path and also showed that minimizing the bandwidth for a tree is NP–complete. With respect to profile storage the results ... |

74 |
The np-completeness of the bandwidth minimization problem
- Papadimitriou
- 1976
(Show Context)
Citation Context ... graph that is NP–hard, except in the trivial case of a path for which the incidence matrix is tridiagonal. The NP–completeness of the bandwidth minimization problem was first proved by Papadimitriou =-=[23]-=-. Gary and Johnson [11] gave a linear time algorithm for finding the minimum bandwidth ordering of a path and also showed that minimizing the bandwidth for a tree is NP–complete. With respect to profi... |

73 |
1976], An Algorithm for Reducing the Bandwidth and Profile of a Sparse Matrix
- Gibbs, Poole, et al.
- 1976
(Show Context)
Citation Context ...f the Cuthill and McKee ordering and the Reversed Cuthill and McKee ordering may depend strongly on the choice of initial node. An iterative procedure has been proposed by Gibbs, Poole and Stockmeyer =-=[14]-=-. The procedure continues after the first level structure has been created by considering all nodes in the final level as an alternate starting node. The node that generates a new level structure with... |

65 |
The theory of graphs and its applications
- Berge
- 1962
(Show Context)
Citation Context ...nimizes fill–in for an arbitrary graph is NP–complete [26, 25]. The type of graph for which there exist an ordering that generates no fill–in during elimination is known as perfect elimination graphs =-=[16, 1, 2, 5, 29, 15]-=-. Determining whether or not a graph is a perfect elimination graph can be made in time O(NM) [25]. The fill–in once an ordering is found can also be computed in time O(NM). But, finding the minimum f... |

41 |
J.G.L.: Fortran at ten gigaflops: the connection machine convolution compiler
- Bromley, Heller, et al.
- 1991
(Show Context)
Citation Context ...special case where the interaction matrix has scalar entries as opposed to block entries, the computations can be highly optimized. It is possible to create special convolution (or stencil) compilers =-=[3]-=-. But, in the case with block entries, the required operations for the interaction between a pair of grid points are no longer simple multiplications and additions, but matrix–vector or matrix–matrix ... |

40 |
Computer implementation of the finite element method
- George
- 1971
(Show Context)
Citation Context ...metric graphs. For an additional discussion of minimum degree orderings and their implementation, see [7]. The Cuthill and McKee ordering [4] and the Reverse Cuthill and McKee ordering used by George =-=[13]-=-. The basic idea in the Cuthill and McKee ordering is to build a level structure defined by picking a starting node for level one. The neighbors of this node form level two. The neighbors of the nodes... |

34 |
Les problèmes de coloration en théorie des graphes
- Berge
- 1960
(Show Context)
Citation Context ...nimizes fill–in for an arbitrary graph is NP–complete [26, 25]. The type of graph for which there exist an ordering that generates no fill–in during elimination is known as perfect elimination graphs =-=[16, 1, 2, 5, 29, 15]-=-. Determining whether or not a graph is a perfect elimination graph can be made in time O(NM) [25]. The fill–in once an ordering is found can also be computed in time O(NM). But, finding the minimum f... |

24 |
Linear algorithms to recognize interval graphs and test for the consecutive ones property
- Booth, Lueker
- 1975
(Show Context)
Citation Context ...NP–complete. With respect to profile storage the results are no better, except for graphs that have an ordering resulting in a dense profile. Such graphs are known as interval graphs. Lueker and Both =-=[21]-=- has devised an O(N + M) algorithm for testing whether or not a graph is an interval graph, where N is the number of nodes and M is the number of edges in the graph. The algorithm is constructive, so ... |

21 |
Complexity bounds for regular finite difference and finite element grids
- Hoffman, Martin, et al.
- 1973
(Show Context)
Citation Context ... nested dissection [12, 27] has been proven to be optimal [19, 20]. It has been shown that any bandwidth or profile ordering for N × N in two dimensions require O(N 3 ) storage and O(N 4 ) operations =-=[22, 27]-=-. In three dimensions, any band or profile ordering for a N × N × N grid requires O(N 6 ) storage and O(N 7 ) work [10]. Table 1 summarizes the these results and give the optimal storage and work requ... |

17 |
Matrix Computation for Engineers and Scientists
- Jennings
- 1977
(Show Context)
Citation Context ...re 16 is created. Now, the matrix structure is ⎛ ⎞ x x x x x x ⎜ ⎟ ⎝ x x x x x⎠ x x and the elimination can be performed with no fill–in. The reason for the much reduced fill– in has been analyzed in =-=[17]-=-. The reversed Cuthill and McKee ordering has been found very effective for a variety of graphs [13] and in subsequent work. The result of the Cuthill and McKee ordering and the Reversed Cuthill and M... |

17 |
The use of linear graphs in Gaussian elimination
- Parter
- 1961
(Show Context)
Citation Context ...imination of all nonzero elements below the diagonal in the pivot column (assuming diagonal pivoting), can be viewed as eliminating the node in the graph that represents the variable to be eliminated =-=[24]-=-. When a node is eliminated from the graph, then edges are inserted between all the nodes adjacent to the eliminated node. A clique is formed between of the nodes adjacent to the eliminated node. To s... |

14 |
Über die Auflösung von Graphen in vollständige Teilgraphen
- Hajnal, Surányi
- 1958
(Show Context)
Citation Context |

12 |
Graph theory and Gaussian elimination
- Tarjan
- 1976
(Show Context)
Citation Context |

9 |
Applications of an element model for gaussian elimination
- Eisenstat, Schultz, et al.
- 1976
(Show Context)
Citation Context ...g for N × N in two dimensions require O(N 3 ) storage and O(N 4 ) operations [22, 27]. In three dimensions, any band or profile ordering for a N × N × N grid requires O(N 6 ) storage and O(N 7 ) work =-=[10]-=-. Table 1 summarizes the these results and give the optimal storage and work requirements for nested dissection, as shown in [22, 10, 28]. 1.3.2 Householder transformations and Givens rotations Pivoti... |

8 |
Models of Massively Parallel Computation, chapter 4
- Johnsson
- 1990
(Show Context)
Citation Context ...s the grid point variables. Communication is only required when the interacting grid points are allocated to different nodes. Using a consecutive allocation scheme with suitably chosen subgrid shapes =-=[18]-=- minimizes the communication (the surface area for an equal amount of communication along all axes). The required data interaction between the grid points, corresponds to a stencil. Thus, it is possib... |

8 |
Algorithmic aspects of vertex elimination directed graphs
- Rose, Tarjan
- 1978
(Show Context)
Citation Context .... Similarly, the graph in Figure 11 also had an elimination order that resulted in no fill–in. Unfortunately, finding an elimination order that minimizes fill–in for an arbitrary graph is NP–complete =-=[26, 25]-=-. The type of graph for which there exist an ordering that generates no fill–in during elimination is known as perfect elimination graphs [16, 1, 2, 5, 29, 15]. Determining whether or not a graph is a... |

6 |
On George’s nested dissection method
- Duff, Erisman, et al.
- 1976
(Show Context)
Citation Context ...th tie breaking is not important. However, for most graphs the tie breaking strategy may have a significant impact on both the total storage and work required. For grid graphs experiments reported in =-=[6]-=- should a difference of about 50% by selecting the first node of minimum degree, where first refers to an initial ordering either in “spiral order” or row major order. In the spiral order nodes are la... |

6 | Node orderings and concurrency for structurallysymmetric sparse problems - Duff, Johnsson - 1989 |

5 |
A recursive analysis of dissection strategies
- Rose, Whitten
- 1976
(Show Context)
Citation Context ...rdering for a N × N × N grid requires O(N 6 ) storage and O(N 7 ) work [10]. Table 1 summarizes the these results and give the optimal storage and work requirements for nested dissection, as shown in =-=[22, 10, 28]-=-. 1.3.2 Householder transformations and Givens rotations Pivoting, or the use of Householder transformations, increase the fill–in. In Householder transformations, a pivot row is computed as a linear ... |

3 |
On the Reduction of Sparse Matrices to Condensed Forms by Similarity Transformations
- Duff, Reid
- 1975
(Show Context)
Citation Context ...arest neighbors, and the fill–in is usually significantly less. The fill–in for elimination using Givens rotations is less than for Householder transformations, but more than for Gaussian elimination =-=[9]-=-. To see these relationships we first compare Givens rotations with Gaussian elimination. Using Givens rotations both the row subject to the elimination and the row used for the elimination are update... |

1 |
Algorithmic Graph Theorey and Perfect Graphs
- Golumbic
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