We give an embedding of the Poincar\'e halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n \cdot \lambda_k(n)$ edges, where $\lambda_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we present a data structure for approximate near-neighbor searching in $H^D$, with construction time $2^{O(D)}n\log n$, query time $2^{O(D)}\log n$ and additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log n$ time.

翻译：我们给出了基于$H^D$二元平铺的庞加莱半空间$H^D$到离散度量空间的嵌入，加性畸变为$O(\log D)$。由此得到以下结果：任意$n$个点构成的子集$P \subset H^D$可嵌入到具有$2^{O(D)}n$个顶点和边的图度量中，加性畸变$O(\log D)$。进一步证明，对任意$k$，可构造$P$的$O(k\log D)$纯加性扩展子图，包含$2^{O(D)}n$个斯坦纳顶点和$2^{O(D)}n \cdot \lambda_k(n)$条边，其中$\lambda_k(n)$为第$k$行逆阿克曼函数。最后，我们提出$H^D$中近似近邻搜索的数据结构，构建时间$2^{O(D)}n\log n$，查询时间$2^{O(D)}\log n$，加性误差$O(\log D)$。上述构造均可在$2^{O(D)}n \log n$时间内完成。