High-dimensional approximate r-nets

Georgia Avarikioti (1) , Ioannis Z. Emiris (1, 2) , Loukas Kavouras (1) , Ioannis Psarros (1, 3)

1 NKUA - National and Kapodistrian University of Athens
2 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
3 CRISAM - Inria Sophia Antipolis - Méditerranée

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. [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 [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.

Mots clés: clustering; nets; approximation; randomized; Metric geometry; High dimension; Approximation algorithms; r-nets; Locality-sensitive hashing

Domaines: Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]; Complexité [cs.CC]

Georgia Avarikioti, Ioannis Z. Emiris, Loukas Kavouras, Ioannis Psarros. High-dimensional approximate r-nets. SODA: ACM/SIAM Symposium on Discrete Algorithms, Jan 2017, Barcelone, Spain. ⟨hal-01636792⟩