Miller, Teng, Thurston, and Vavasis proved that every knearest neighbor graph (k-NNG) in R has 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 R is at O( 1 n2/d ). In this paper, we extend these two results to nearest neighbor graphs in a metric space w...

We study k-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least (1 − δ), simultaneously has a cost of at most (1 + ǫ) times the optimal cost and ...

This paper presents novel techniques that allow the solution to several open problems regarding embedding of finite metric spaces into Lp. We focus on proving near optimal bounds on the dimension with which arbitrary metric spaces embed into Lp. The dimension of the embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no ...

Point location problems for n points in d-dimensional Euclidean space (and `p spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than d · n log n time, or (Barely-Subquadratic) f(d) ·n2−1/Θ(d) time, for various f . For small d and large n, “nearly-linear” running times are generally feasible, while the “barely-subquadratic” times are generally in...

We show that for every set S of n points in the plane and a designated point rt ∈ S, there exists atree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distancebetween rt and v in T is within a factor of (1+2) close to their Euclidean distance ‖rt, v‖. We call thesetrees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction a...

Point location problems for n points in d-dimensional Euclidean space (and lp spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than d ·n log n time, or (Barely-Subquadratic) f (d) ·n2−1/Θ(d) time, for various functions f . For small d and large n, “nearly-linear” running times are generally feasible, while the “barelysubquadratic” times are gen...

A metric space has doubling dimension d if for every ρ > 0, every ball of radius ρ can be covered by at most 2d balls of radius ρ/2. This generalizes the Euclidean dimension, because the doubling dimension of Euclidean space Rd is proportional to d. The following results are shown, for any d ≥ 1 and any metric space of size n and doubling dimension d: First, the maximum number of diametral pair...