## Section: New Results

### High-dimensional approximate $r$-nets

Participants : Ioannis Emiris, Ioannis Psarros.

The construction of $r$-nets offers a powerful tool in computational and metric geometry. In [17], 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 $\u03f5>0$, the approximation factor is $1+\u03f5$ 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 is improved in terms of complexity by reducing the dependence on $\u03f5$, provided that $\u03f5$ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's (2015) 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 $(1+\u03f5)$-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

Joint with G. Avarikioti, L. Kavouras.