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

We give an embedding of the Poincar\'e 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 present a data structure for approximate near-neighbor searching in HD, with construction time 2O(D)nlog n, query time 2O(D)log n and additive error O(log D). These constructions can be done in 2O(D)n log n time.