Combinatorial and Spectral Aspects of Nearest Neighbor Graphs in Doubling Dimensional and Nearly-Euclidean Spaces

Miller, Teng, Thurston, and Vavasis proved that every knearest neighbor graph (k-NNG) in R has a balanced vertex separator of size O(n1−1/dk1/d). Later, Spielman and Teng proved that the Fiedler value — the second smallest eigenvalue of the graph — of the Laplacian matrix of a k-NNG in R is at O( 1 n2/d ). In this paper, we extend these two results to nearest neighbor graphs in a metric space with doubling dimension γ and in nearly-Euclidean spaces. We prove that for every l > 0, each k-NNG in a metric space with doubling dimension γ has a vertex separator of size O(kl(32l + 8) log L S log n + n l ), where L and S are respectively the maximum and minimum distances between any two points in P . We show how to use the singular value decomposition method to approximate a k-NNG in a nearly Euclidean space by an Euclidean k-NNG. This approximation enables us to obtain an upper bound on the Fiedler value of the k-NNG in a nearly Euclidean space. keywords Doubling dimension, shallow minor, neighborhood system, metric embedding, Fiedler value. ? Part of this work was done while visiting Computer Science Department at Boston University. In part supported by the National Grand Fundamental Research 973 Program of China under Grant (2004CB318108, 2004CB318110, 2003CB317007), the National Natural Science Foundation of China Grant (60553001, 60321002) and the National Basic Research Program of China Grant (2007CB807900, 2007CB807901). ?? Part of this work was done while visiting Tsinghua University and Microsoft Research Asia Lab.