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

Abstract

Miller, Teng, Thurston, and Vavasis proved a geometric separator theorem which implies that the k -nearest neighbor graph ( k -NNG) of every set of n points in R d has a balanced vertex separator of size O ( n 1 − 1 / d k 1 / d ) . Spielman and Teng then proved that the Fiedler value — the second smallest eigenvalue of the Laplacian matrix — of the k -NNG of any n points in R d is O ( ( k / n ) 2 / d ) . In this paper, we extend these two results to nearest neighbor graphs in a metric space with a finite doubling dimension and in a metric space that is nearly-Euclidean. We prove that for every l > 0 , if ( X , dist ) forms a metric space with doubling dimension γ , then the k -NNG of every set P of n points in X has a vertex separator of size O ( k 2 l ( 64 l + 8 ) 2 γ log 2 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 a Euclidean k -NNG. This approximation enables us to obtain an upper bound on the Fiedler value of k -NNGs in a nearly-Euclidean space.