High-Dimensional Approximate r-Nets

Abstract

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed $\epsilon >0$, the approximation factor is $1+\epsilon$ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555) in terms of complexity, by reducing the dependence on $\epsilon$, provided that $\epsilon$ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the $(1+\epsilon )$-approximate k-th nearest neighbor distance in time subquadratic in the size of the input.