Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
Qualitative analysis of distributed physical systems with applications to control synthesis
AAAI '98/IAAI '98 Proceedings of the fifteenth national/tenth conference on Artificial intelligence/Innovative applications of artificial intelligence
The Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of a Laplacian
Combinatorics, Probability and Computing
The Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of a Laplacian
Combinatorics, Probability and Computing
Isoperimetric Graph Partitioning for Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
ICTIR '09 Proceedings of the 2nd International Conference on Theory of Information Retrieval: Advances in Information Retrieval Theory
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Spectral partitioning methods use the Fiedler vector---the eigenvector of the second-smallest eigenvalue of the Laplacian matrix---to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on bounded-degree planar graphs and finite element meshes--- the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is $\O{\sqrt{n}}$ for bounded-degree planar graphs and two-dimensional meshes and $\O{n^{1/d}}$ for well-shaped $d$-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs.