High-dimensional approximate r-nets

Georgia Avarikioti, Ioannis Z. Emiris, Loukas Kavouras, Ioannis Psarros

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 ∊ > 0, the approximation factor is 1 + ∊ 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. [EHS15] in terms of complexity, by reducing the dependence on ∊, provided that ∊ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] 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 + ∊)-approximate kth nearest neighbor distance in time subquadratic in the size of the input.