High-Dimensional Approximate r-Nets

摘要

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>0the 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 ) 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. 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.