Abstract
In this paper we present new algorithms for spectral graph partitioning. Previously, the best partitioning methods were based on a combination of Combinatorial algorithms and application of the Lanczos method when the graph allows this method to be cheap enough. Our new algorithms are purely spectral. They calculate the Fiedler vector of the original graph and use the information about the problem in the form of a preconditioner for the graph Laplacian. In addition, we use a favorable subspace for starting the Davidson algorithm and reordering of variables for locality of memory references.
This research was supported by NSF grant ASC-9528912, and currently by NSF/DARPA DMS-9874015
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S. Barnard and H. Simon. A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. In Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, Norfolk, Virginia, 1993. SIAM, SIAM.
L. Borges and S. Oliveira. A parallel Davidson-type algorithm for several eigenvalues. Journal of Computational Physics, (144):763–770, August 1998.
T. Chan, P. Ciarlet Jr., and W. K. Szeto. On the optimality of the median cut spectral bisection graph partitioning method. SIAM Journal on Computing, 18(3):943–948, 1997.
E. Davidson. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. Journal of Computational Physics, 17:87–94, 1975.
M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 23:298–305, 1973.
M. Fiedler. A property of eigenvectors of non-negative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal, 25:619–632, 1975.
S. Guattery and G. L. Miller. On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications, 19:701–719, 1998.
M. Holzrichter and S. Oliveira. New spectral graph partitioning algorithms. submitted.
M. Holzrichter and S. Oliveira. New graph partitioning algorithms. 1998. The University of Iowa TR-120.
G. Karypis and V. Kumar. Multilevel k-way partitioning scheme for irregular garphs. to appear in the Journal of Parallel and Distributed Computing.
G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1999.
B. Kernighan and S. Lin. An efficient heuristic procedure for partitioning graphs. The Bell System Technical Journal, 49:291–307, February 1970.
S. McCormick. Multilevel Adaptive Methods for Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1989.
S. Oliveira. A convergence proof of an iterative subspace method for eigenvalues problem. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics Selected Papers, pages 316–325. Springer, January 1997.
Alex Pothen, Horst D. Simon, and Kang-Pu Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl., 11(3):430–452, 1990. Sparse matrices (Gleneden Beach, OR, 1989).
H. D. Simon and S. H. Teng. How good is recursive bisection. SIAM Journal on Scientific Computing, 18(5):1436–1445, July 1997.
D. Spielman and S. H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium Foundations of Computer Science, Burlington, Vermont, October 1996. IEEE, IEEE Press.
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Holzrichter, M., Oliveira, S. (1999). A graph based method for generating the fiedler vector of irregular problems. In: Rolim, J., et al. Parallel and Distributed Processing. IPPS 1999. Lecture Notes in Computer Science, vol 1586. Springer, Berlin, Heidelberg . https://doi.org/10.1007/BFb0097982
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DOI: https://doi.org/10.1007/BFb0097982
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