We construct coresets for the continuous $k$-center problem in fixed-dimensional hyperbolic space $\mathbb H^D$. The input is a set $P$ of $n$ points in $\mathbb H^D$, where $D=O(1)$, and the centers may be placed anywhere in the ambient hyperbolic space. Given $\varepsilon\in(0,1)$, we construct a subset $P_\varepsilon\subseteq P$ such that every optimal continuous $k$-center solution for $P_\varepsilon$ is a $(1+\varepsilon)$-approximation for $P$. The main difficulty is the exponential volume growth of hyperbolic balls, which prevents a direct grid-based coreset from having size independent of the input radius. We overcome this by dividing the construction according to the farthest-first scale. At bounded scales, we use local Euclidean grids in the Poincaré ball model. At intermediate scales, we use an anchor-centered shell--cone decomposition together with exact distance profiles obtained from the hyperbolic law of cosines. At large scales, we avoid discretizing the ambient ball and instead keep input witnesses indexed by coarse profiles of the induced $k$-center distance functions on each shell--cone bucket. The resulting coreset has size $\left(1/\varepsilon\right)^{O(kD)}$ and can be constructed in time $O(nk\left(1/\varepsilon\right)^{O(kD)}).$ Both bounds are independent of the input radius, and the coreset size is also independent of $n$. Consequently, for fixed $D$, $k$, and $\varepsilon$, this gives a linear-time construction of a constant-size coreset for the continuous $k$-center problem in hyperbolic space.

翻译：我们针对固定维双曲空间$\mathbb H^D$中的连续$k$-中心问题构建核心集。输入为$\mathbb H^D$中的$n$个点构成的集合$P$（其中$D=O(1)$），中心点可置于双曲空间中任意位置。给定$\varepsilon\in(0,1)$，我们构造子集$P_\varepsilon\subseteq P$，使得$P_\varepsilon$的任意最优连续$k$-中心解均是$P$的$(1+\varepsilon)$-近似解。主要困难在于双曲球的体积呈指数增长，这使得基于网格的直接核心集规模无法独立于输入半径。我们通过按最远优先尺度划分构造来克服这一困难：在有限尺度下，使用庞加莱球模型中的局部欧几里得网格；在中间尺度下，采用锚点中心的壳-锥分解并结合双曲余弦定理获得的精确距离剖面；在大尺度下，避免对双曲球离散化，转而通过每个壳-锥桶上诱导的$k$-中心距离函数的粗粒度剖面来保留输入点证据。最终核心集规模为$\left(1/\varepsilon\right)^{O(kD)}$，可在$O(nk\left(1/\varepsilon\right)^{O(kD)})$时间内构建。两个边界均与输入半径无关，核心集规模亦与$n$无关。因此，对于固定$D$、$k$和$\varepsilon$，该算法实现了双曲空间中连续$k$-中心问题的线性时间常数规模核心集构建。