## A spectral algorithm for envelope reduction of sparse matrices (1993)

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Venue: | ACM/IEEE CONFERENCE ON SUPERCOMPUTING |

Citations: | 60 - 5 self |

### BibTeX

@INPROCEEDINGS{Barnard93aspectral,

author = {Stephen T. Barnard and Alex Pothen and Horst D. Simon},

title = {A spectral algorithm for envelope reduction of sparse matrices},

booktitle = {ACM/IEEE CONFERENCE ON SUPERCOMPUTING},

year = {1993},

pages = {493--502},

publisher = {ACM}

}

### Years of Citing Articles

### OpenURL

### Abstract

The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelope-reducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. This Laplacian eigenvector solves a continuous relaxation of a discrete problem related to envelope minimization called the minimum 2-sum problem. The permutation vector computed by the spectral algorithm is a closest permutation vector to the specified Laplacian eigenvector. Numerical results show that the new reordering algorithm usually computes smaller envelope sizes than those obtained from the current standards such as the Gibbs-Poole-Stockmeyer (GPS) algorithm or the reverse Cuthill-McKee (RCM) algorithm in SPARSPAK, in some cases reducing the envelope by more than a factor of two.

### Citations

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Citation Context ...algorithm for computing a few eigenvalues and eigenvectors of large sparse symmetric matrices is the Lanczos algorithm. Since the Lanczos algorithm is discussed extensively in the textbook literature =-=[16, 26]-=-, we do not include a detailed description of the standard algorithm here. Recently, we have developed a much more efficient multilevel method for finding a second eigenvector [3]. The multilevel meth... |

550 |
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Citation Context ...utations. Ordering algorithms such as minimum-degree and nested dissection have been developed for reducing fill in direct methods for solving sparse, symmetric positive definite systems of equations =-=[7, 12, 27]-=-. Various ordering algorithms for reducing the envelope (variable band or profile) of sparse matrices, such as the reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), and Gibbs-King (GK) algori... |

512 |
Partitioning Sparse Matrices with Eigenvectors of Graphs
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Citation Context ...sed a second eigenvector of the Laplacian matrix for computing a spectral nested dissection ordering, and for partitioning computations on finite element meshes on a distributed-memory multiprocessor =-=[29, 30, 31]-=-. The eigenvector of the adjacency matrix corresponding to the largest eigenvalue has been used to find a pseudoperipheral node by Grimes et al. [17]. A companion paper [13] provides theoretical justi... |

512 | Computer Solution of Large Sparse Positive Definite Systems. Englewood Cliffs - George, Liu - 1981 |

430 |
Algebraic connectivity of graphs
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Citation Context ...of the components of a second Laplacian eigenvector is shown to yield a feasible solution to the 2-sum problem that is closest to an infeasible solution for which the lower bound is attained. Fiedler =-=[9, 11]-=- studied the properties of the second Laplacian eigenvalue and a corresponding eigenvector and their relationship to the connectivity of a graph, and also observed [10] that the differences in the com... |

299 |
Partitioning of unstructured problems for parallel processing
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Citation Context ...sed a second eigenvector of the Laplacian matrix for computing a spectral nested dissection ordering, and for partitioning computations on finite element meshes on a distributed-memory multiprocessor =-=[29, 30, 31]-=-. The eigenvector of the adjacency matrix corresponding to the largest eigenvalue has been used to find a pseudoperipheral node by Grimes et al. [17]. A companion paper [13] provides theoretical justi... |

278 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems
- Barnard, Simon
- 1993
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Citation Context ...muting the components of a second Laplacian eigenvector in nonincreasing (or nondecreasing) order. For large matrices, the eigenvector computation is performed by a 'multilevel' approach described in =-=[3]-=-. Earlier, we had used a second eigenvector of the Laplacian matrix for computing a spectral nested dissection ordering, and for partitioning computations on finite element meshes on a distributed-mem... |

237 |
A property of eigenvectors of non-negative symmetric matrices and its applications to graph theory
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Citation Context ...of the components of a second Laplacian eigenvector is shown to yield a feasible solution to the 2-sum problem that is closest to an infeasible solution for which the lower bound is attained. Fiedler =-=[9, 11]-=- studied the properties of the second Laplacian eigenvalue and a corresponding eigenvector and their relationship to the connectivity of a graph, and also observed [10] that the differences in the com... |

220 |
Reducing the bandwidth of sparse symmetric matrices
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Citation Context ...cian spectra to combinatorial problems. The spectral envelope-reduction algorithm has several features which set it apart from the earlier reordering algorithms such as the GPS, GK, or RCM algorithms =-=[5, 12, 15, 21]-=-. These algorithms employ local-search in the adjacency graph of the matrix. All of them try to find a pseudo-diameter in the graph by generating a long level-structure by breadth-first-search beginni... |

85 |
The effect of ordering on preconditioning conjugate gradients
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Citation Context ...ears in several other applications. The RCM ordering has been found to be an effective preordering in computing incomplete factorization preconditioners for preconditioned conjugate gradients methods =-=[6, 8]-=-. Such orderings have also been used in parallel matrix-vector multiplication and tridiagonalization of sparse symmetric matrices. The wider applicability of envelope-reducing orderings justifies a fr... |

75 |
An Algorithm for Reducing the Bandwidth and Profile of Sparse Matrix
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Citation Context ...e method of choice for solving large-scale systems of linear equations in many structural engineering applications, for example in the computational structural mechanics testbed (CSM) at NASA Langley =-=[20]-=-. Implementations of these methods are also widely distributed in most of the finite element software packages such as MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization ... |

55 | Optimal linear labelings and eigenvalues of graphs - JUVAN, MOHAR - 1992 |

53 |
Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems
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Citation Context ...ears in several other applications. The RCM ordering has been found to be an effective preordering in computing incomplete factorization preconditioners for preconditioned conjugate gradients methods =-=[6, 8]-=-. Such orderings have also been used in parallel matrix-vector multiplication and tridiagonalization of sparse symmetric matrices. The wider applicability of envelope-reducing orderings justifies a fr... |

50 |
Vectorization of a multiprocessor multifrontal code
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Citation Context ... MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization of a matrix in envelope format have been investigated [28, 33]. Efficient implementations of sparse matrix algorithms =-=[1, 2, 22, 32]-=- on supercomputers demonstrate that very high levels of performance are attainable with general sparse algorithms. Hence there are no good reasons to use envelope schemes for sparse matrix factorizati... |

50 | Computer Solution of Large Sparse Positive De nite Systems - George, Liu - 1981 |

48 | Comparative analysis for the Cuthill-McKee and the reverse CuthillÂ·McKee ordering algorithms for sparse matrices - Liu, Sherman - 1976 |

45 |
Recent progress in sparse matrix methods for large linear systems
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Citation Context ... MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization of a matrix in envelope format have been investigated [28, 33]. Efficient implementations of sparse matrix algorithms =-=[1, 2, 22, 32]-=- on supercomputers demonstrate that very high levels of performance are attainable with general sparse algorithms. Hence there are no good reasons to use envelope schemes for sparse matrix factorizati... |

43 | Eigenvalues in combinatorial optimization
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(Show Context)
Citation Context ...tor is an approximate measure of the distance between the vertices. Juvan and Mohar [19] havesadvocated the use of this eigenvector to compute bandwidth and p-sum reducing orderings. Mohar and Poljak =-=[25]-=- have recently provided a comprehensive survey of the applications of Laplacian spectra to combinatorial problems. The spectral envelope-reduction algorithm has several features which set it apart fro... |

30 | Spectral nested dissection
- Pothen, Simon, et al.
- 1992
(Show Context)
Citation Context ...sed a second eigenvector of the Laplacian matrix for computing a spectral nested dissection ordering, and for partitioning computations on finite element meshes on a distributed-memory multiprocessor =-=[29, 30, 31]-=-. The eigenvector of the adjacency matrix corresponding to the largest eigenvalue has been used to find a pseudoperipheral node by Grimes et al. [17]. A companion paper [13] provides theoretical justi... |

25 | Meurant, The effect of orderings on preconditioned conjugate gradients - Duff, A - 1989 |

22 |
Implementation of the Gibbs-Poole-Stockmeyer and Gibbs-King Algorithms
- Lewis
- 1982
(Show Context)
Citation Context ...for reducing the envelope (variable band or profile) of sparse matrices, such as the reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), and Gibbs-King (GK) algorithms, have also been designed =-=[12, 15, 21]-=-. Although envelope-reducing orderings were developed for use in envelope schemes for direct factorization, these orderings have been used in the past few years in several other applications. The RCM ... |

20 | Sparse Matrix - Pissanetzky - 1984 |

15 |
A generalized envelope method for sparse factorization by rows
- Liu
- 1991
(Show Context)
Citation Context ...monstrating the efficiency of the new spectral algorithm offers potential performance improvements in these packages without making substantial changes to the underlying data structures. Further, Liu =-=[23]-=- has described a generalized envelope algorithm for computing the numerical factorization by rows, and his results show that such a scheme can compete with general sparse algorithms. The following is ... |

12 | A spectral approach to bandwidth and separator problems in graphs - HELMBERG, MOHAR, et al. - 1993 |

8 |
Algorithm 509: A hybrid profile reduction algorithm
- Gibbs
- 1976
(Show Context)
Citation Context ...th. This is the reason why the latter two algorithms require more time than the RCM algorithm. Generally the GPS algorithm yields a lower bandwidth while the GK algorithm yields a lower envelope size =-=[14, 21]-=-. Our results are in agreement with this conclusion. It should be pointed out that n - 2680 was the largest order of the problems considered in earlier work, and that the results reported here are for... |

7 | Performance of a supernodal general sparse solver on the CRAY Y-MP: 1.68 GFLOPS with autotasking - Simon, Vu, et al. - 1989 |

6 |
Parallel-Vector Solution of Large-Scale Structural Analysis Problems
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Citation Context ...tributed in most of the finite element software packages such as MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization of a matrix in envelope format have been investigated =-=[28, 33]-=-. Efficient implementations of sparse matrix algorithms [1, 2, 22, 32] on supercomputers demonstrate that very high levels of performance are attainable with general sparse algorithms. Hence there are... |

6 | An algorithm for reducing the bandwidth and pro le of a sparse matrix - Gibbs, Poole, et al. - 1976 |

4 |
The Impact of Hardware Gather/Scatter On Sparse Gaussian Elimination
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- 1988
(Show Context)
Citation Context ... MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization of a matrix in envelope format have been investigated [28, 33]. Efficient implementations of sparse matrix algorithms =-=[1, 2, 22, 32]-=- on supercomputers demonstrate that very high levels of performance are attainable with general sparse algorithms. Hence there are no good reasons to use envelope schemes for sparse matrix factorizati... |

4 | A new algorithm for finding a pseudoperipheral node in a graph - Grimes, Pierce, et al. - 1990 |

3 |
An analysis of the spectral approach to envelope reduction via quadratic assignment problems
- George, Pothen
- 1993
(Show Context)
Citation Context ...ory multiprocessor [29, 30, 31]. The eigenvector of the adjacency matrix corresponding to the largest eigenvalue has been used to find a pseudoperipheral node by Grimes et al. [17]. A companion paper =-=[13]-=- provides theoretical justification for the spectral envelopereduction algorithm by considering a closely related problem called the 2-sum problem. (This problem is defined in the next section.) It is... |

3 |
The solution of linear systems of equations with a structural analysis code
- Poole, Overman
- 1988
(Show Context)
Citation Context ...tributed in most of the finite element software packages such as MSC/NASTRAN or ANSYS. Parallel algorithms for the actual numerical factorization of a matrix in envelope format have been investigated =-=[28, 33]-=-. Efficient implementations of sparse matrix algorithms [1, 2, 22, 32] on supercomputers demonstrate that very high levels of performance are attainable with general sparse algorithms. Hence there are... |

3 | On the near optimality of the recursive spectral bisection method for graph partitioning - Chan, Szeto - 1993 |

2 | Algebraische zusammenhangszahl der graphen und ihre numerische bedeutung, in Numerische Methoden bei graphentheoretischen und kombinatorischen - Fiedler - 1974 |

2 | Algorithm 509: A hybrid pro le reduction algorithm - Gibbs - 1976 |

2 | A new algorithm for nding a pseudoperipheral nodeinagraph - Grimes, Pierce, et al. - 1990 |

1 |
On the near optimality of the recnrsive spectral bisection method for graph partitioning, manuscript
- CHAN, SZETO
- 1993
(Show Context)
Citation Context ...s the continuous approximation to the 2-sum problem. Now we prove that a permutation vector p_,_ induced by x__ is a closest vector in _ to x2. Earlier a similar result was obtained by Chan and Szeto =-=[4]-=- for the graph bisection problem. THEOREM 2.3. The vector P--minduced by a second Laplacian eigenvector x 2 is a closest (in the 2-norm) permutation vector to x 2. In other words, = arg minll p - x__l... |

1 |
Aigebraische zusammenhangszahl der graphen nnd ihre numerische bedeutung, in Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen, Oberwolfach
- FmDLER
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(Show Context)
Citation Context ...r bound is attained. Fiedler [9, 11] studied the properties of the second Laplacian eigenvalue and a corresponding eigenvector and their relationship to the connectivity of a graph, and also observed =-=[10]-=- that the differences in the components of this eigenvector is an approximate measure of the distance between the vertices. Juvan and Mohar [19] havesadvocated the use of this eigenvector to compute b... |

1 |
Computer Solution of Large Sparse Positive Definite Systems
- GEORQE
- 1981
(Show Context)
Citation Context ...utations. Ordering algorithms such as minimum-degree and nested dissection have been developed for reducing fill in direct methods for solving sparse, symmetric positive definite systems of equations =-=[7, 12, 27]-=-. Various ordering algorithms for reducing the envelope (variable band or profile) of sparse matrices, such as the reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), and Gibbs-King (GK) algori... |

1 |
Comparative analysis of the CuthilI-Mckee and the reverse Cuthiil-Mckee ordering algorithms for sparse matrices
- LIV, SHERMAN
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(Show Context)
Citation Context ...tices in Y \ Y that are adjacent to some vertex in Y. We will say that an ordering is an adjacency ordering if Vj+I E adj(Vj), for j = 1,..., n - 1.sThe size ladj(_)l hasbeencalled the jth frontwidth =-=[24]-=-, and corresponds to the size of the j-th column of the envelope of A. Hence an alternative expression for the the envelope size is Esize(A) = _ ladj(V_)l. This expression for the envelope size shows ... |

1 |
Sparse Matriz
- PISSANETZKY
- 1984
(Show Context)
Citation Context ...utations. Ordering algorithms such as minimum-degree and nested dissection have been developed for reducing fill in direct methods for solving sparse, symmetric positive definite systems of equations =-=[7, 12, 27]-=-. Various ordering algorithms for reducing the envelope (variable band or profile) of sparse matrices, such as the reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), and Gibbs-King (GK) algori... |

1 |
Performance ofasupernodal general sparse solver on
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Citation Context |

1 | CSM testbed development and large-scale structural applications - Knight, Gillian, et al. |