High-dimensional approximate $r$-nets

TL;DR

This paper introduces a new randomized algorithm for efficiently computing approximate r-nets in high-dimensional Euclidean spaces, improving complexity over previous LSH-based methods and enabling faster solutions to geometric problems like approximate nearest neighbor search.

Contribution

The paper presents a novel randomized algorithm for approximate r-nets that reduces complexity and does not rely on LSH, improving efficiency in high-dimensional geometry computations.

Findings

Algorithm achieves (1+ε)-approximate r-nets with polynomial dependence on dimension.

Complexity is subquadratic in the number of points, faster than previous methods.

Enables efficient approximate solutions to high-dimensional geometric problems.

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. More specifically, the best previously known LSH-based construction of Eppstein et al.\ \cite{EHS15} is improved in terms of complexity by reducing the dependence on $\epsilon$, provided that $\epsilon$ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's \cite{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 $(1+\epsilon)$-approximate $k$th nearest neighbor distance in time subquadratic in the size of the input.