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

• Authors: Yingchao Zhao;Shang-hua Teng
• Venue: Theoretical Computer Science
• Year: 2009

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^/^dk^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 l0, if (X,dist) forms a metric space with doubling dimension @c, then the k-NNG of every set P of n points in X has a vertex separator of size O(k^2l(64l+8)^2^@clog^2LSlogn+nl), 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.