Combinatorial and spectral aspects of nearest neighbor graphs in doubling dimensional and nearly-Euclidean spaces

• Authors: Yingchao Zhao;Shang-Hua Teng
• Venue: TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
• Year: 2007

Miller, Teng, Thurston, and Vavasis proved that every k- nearest neighbor graph (k-NNG) in Rdhas 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 Rd 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(k2l(32l + 8)2γ log2 L/S log n + n/l), where L and S are respectively the maximum and minimum distances between any two points in P, and P is the point set that constitutes the metric space. 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.