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

Miller, Teng, Thurston, and Vavasis proved that every k-nearest neighbor graph (k-NNG) in R-d has a balanced vertex separator of size O(n (1 - 1 / d) k (1 / 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-d is at O(n(2)/d/1). In this paper, we extend these two results to nearest neighbor graphs in a metric space with doubling dimension gamma and in nearly-Euclidean spaces. We prove that for every l > 0, each k-NNG in a metric space with doubling dimension gamma has a vertex separator of size O(k(2)l(32l + 8)(2 gamma) log(2) S/L log n + l/n), 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.