Approximation Algorithms in Hyperbolic Spaces with Constant Additive Error

Abstract

Previous work in computational geometry has focused on Euclidean spaces, where the distance between two points p and q is given by the L2 norm of the vector pq. In this thesis, we consider algorithmic problems in hyperbolic spaces, using different metrics. Our motivation is that, even though Euclidean spaces are suitable for many applications, there has been recently some interest in hyperbolic spaces for applications in computer networks, artificial neural networks, and computer vision. In the main part of this thesis, we present an embedding of the D-dimensional Poincaré half-space into a discrete hyperbolic space that is based on a binary tiling of the upper half-space. Based on this embedding, we obtain an embedding of any finite subset of the Poincaré half-space into a graph metric with a linear number of edges, and a constant additive error. We extend this result to obtain a spanner with a linear number of Steiner points, and a linear number of edges, still with a constant additive error. Both of these constructions can be made in near-linear time. Finally, we show how to construct an approximate Voronoi diagram in an hyperbolic space, with constant additive error. It yields a data structure for answering approximate near-neighbor searching queries in logarithmic time, with constant additive error. The approximate Voronoi diagram and the associated data structure can be computed in near-linear time.