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

Abstract. Miller, Teng, Thurston, and Vavasis proved that every knearest 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 ( 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(k 2 l(32l + 8) 2γ 2 L n log log n +), where L and S l 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