z Abstract. A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work in an envelope Cholesky factorization. We formulate the latter two problems as quadratic assignment problems, and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a relaxation of the problem, and then show that the spectral ordering finds a permutation matrix closest to an orthogonal matrix attaining the lower bound. This provides stronger justification of the spectral envelope reduction algorithm than previously known. The lower bound on the 2-sum is seen to be tight for reasonably "uniform" finite element meshes. We show that problems with bounded separator sizes also have bounded envelope parameters. Key words. 1-sum problem, 2-sum problem, envelope reduction, eigenvalues of graphs, Laplacian
|
7709
|
Computers and Intractability: A Guide to the Theory of NP-Completeness
– Garey, Johnson
- 1979
|
|
452
|
Direct Methods for Sparse Matrices
– Duff, Erisman, et al.
- 1986
|
|
430
|
Partitioning sparse matrices with eigenvectors of graphs
– Pothen, Simon, et al.
- 1990
|
|
406
|
Computer Solution of Large Sparse Positive Definite Systems
– George, Liu
- 1981
|
|
300
|
A separator theorem for planar graphs
– Lipton, Tarjan
- 1979
|
|
262
|
Partitioning of unstructured problems for parallel processing
– Simon
- 1991
|
|
199
|
Algebraic connectivity of graphs
– Fiedler
- 1973
|
|
161
|
An improved spectral graph partitioning algorithm for mapping parallel computations. SAND92-1460, Sandia National Laboratories
– Hendrickson, Leland
- 1992
|
|
143
|
Reducing the bandwidth of sparse symmetric matrices
– CUTHILL, McKEE
- 1969
|
|
95
|
Spectral partitioning works: Planar graphs and finite element meshes
– Spielman, Teng
- 1996
|
|
70
|
1, isoperimetric inequalities for graphs and super concentrators
– Alon, Millman
- 1985
|
|
68
|
Graph Algorithms. Computer Science
– Even
- 1979
|
|
64
|
The quadratic assignment problem
– Burkard, Çela, et al.
- 1998
|
|
56
|
An algorithm for reducing the bandwidth and profile of a sparse matrix
– GIBBS, POOLE, et al.
- 1976
|
|
52
|
Ordering methods for preconditioned conjugate gradients methods applied to unstructured grid problems
– D'AZEVEDO, FORSYTH, et al.
- 1992
|
|
50
|
A projection technique for partitioning the nodes of a graph
– RENDL, WOLKOWICZ
- 1995
|
|
47
|
A new lower bound via projection for the quadratic assignment problem
– HADLEY, RENDL, et al.
- 1992
|
|
43
|
H.D.: A spectral algorithm for envelope reduction of sparse matrices
– Barnard, Pothen, et al.
- 1993
|
|
40
|
Comparative analysis of the Cuthill–McKee and the reverse Cuthill–McKee ordering algorithms for sparse matrices
– Liu, Sherman
- 1973
|
|
38
|
Eigenvalues in combinatorial optimization
– MOHAR, POLJAK
- 1992
|
|
33
|
Spectral nested dissection
– POTHEN, SIMON, et al.
- 1992
|
|
28
|
Computer implementation of the finite element method
– GEORGE
- 1971
|
|
23
|
Meurant, The effect of ordering on preconditioned conjugate gradients. BIT
– Duff, A
- 1989
|
|
20
|
Quadratic assignment problems
– Finke, Burkard, et al.
- 1987
|
|
19
|
The use of profile reduction algorithms with a frontal code
– Duff, Reid, et al.
- 1989
|
|
19
|
Implementations of the Gibbs-Poole-Stockmeyer and Gibbs-King algorithms
– Lewis
- 1982
|
|
15
|
An algorithm for profile and wavefront reduction of sparse matrices
– Sloan
- 1986
|
|
14
|
A spectral approach to bandwidth and separator problems in graphs
– Helmberg, Mohar, et al.
- 1993
|
|
11
|
Algorithm 509: A hybrid profile reduction algorithm
– GIBBS
- 1976
|
|
11
|
Laplace eigenvalues and bandwidth-type invariants of graphs
– Juvan, Mohar
- 1993
|
|
9
|
Automatic mesh partitioning, in Graph Theory and Sparse Matrix Computation
– Miller, Teng, et al.
- 1993
|
|
9
|
Node and element resequencing using Laplacian of a finite element graph, Part-I-general concepts and algorithms
– Paulino, Menezes, et al.
|
|
6
|
Dynamic Load Balancing in
– Hendrickson, Devine
|
|
5
|
A new algorithm for finding a pseudoperipheral node in a graph
– Grimes, Pierce, et al.
- 1990
|
|
4
|
Minimum profile of grid networks in structure analysis
– Lin, Yuan
- 1993
|
|
3
|
An improved spectral nested dissection algorithm
– Pothen, Wang
- 1994
|
|
2
|
linear labelings and eigenvalues of graphs
– Optimal
- 1992
|
|
2
|
A refined spectral algorithm to reduce the envelope and wavefront of sparse matrices, BIT
– Kumfert, Pothen
- 1997
|
|
2
|
load balancing
– Multidimensional
- 1993
|
|
