Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces

Abstract

We give an embedding of the Poincaré halfspace HD into a discrete metric space based on a binary tiling of HD, with additive distortion O(log⁡D). It yields the following results. We show that any subset P of n points in HD can be embedded into a graph-metric with 2O(D)n vertices and edges, and with additive distortion O(log⁡D). We also show how to construct, for any k, an O(klog⁡D)-purely additive spanner of P with 2O(D)n Steiner vertices and 2O(D)n⋅λk(n) edges, where λk(n) is the kth-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for P of size 2O(D)n. It allows us to answer approximate near-neighbor queries in 2O(D)+O(Dlog⁡n) time, with additive error O(log⁡D). These constructions can be done in 2O(D)nlog⁡n time. © 2024 Elsevier B.V.