In Euclidean Uniform Facility Location (UFL), the input is a set of clients in $\mathbb{R}^d$ and the goal is to place facilities to serve them, so as to minimize the total cost of opening facilities plus connecting the clients. We study the setting of dynamic geometric streams, where the clients are presented as a sequence of insertions and deletions of points in the grid $\{1,\ldots,\Delta\}^d$, and we focus on the \emph{high-dimensional regime}, where the algorithm must use space polynomial in $d\cdot\log\Delta$. We present a new algorithmic framework, based on importance sampling, for $O(1)$-approximation of UFL using only $\mathrm{poly}(d\cdot\log\Delta)$ space. This framework is easy to implement in two passes, one for sampling points and the other for estimating their contribution. Over random-order streams, we can extend this to one pass by using the two halves of the stream separately. Our main result, for arbitrary-order streams, computes $O(d / \log d)$-approximation in one pass by combining the two passes differently. This improves upon previous algorithms that either need space $\exp(d)$ or only guarantee $O(d\cdot\log^2\Delta)$-approximation, and therefore our algorithms for high dimension are the first to avoid the $O(\log\Delta)$-factor in approximation that is inherent to the widely-used quadtree decomposition. Our improvement is achieved by employing a geometric hashing scheme that maps points in $\mathbb{R}^d$ into buckets of bounded diameter, with the key property that every point set of small-enough diameter is hashed into few buckets. By applying an alternative bound for this hashing, we also obtain an $O(1 / \epsilon)$-approximation in one pass, using larger but still sublinear space $O(n^{\epsilon})$ where $n$ is the number of clients. We complement our results by showing $1.085$-approximation requires space exponential in $\mathrm{poly}(d\cdot\log\Delta)$.

翻译：在欧几里得均匀设施选址问题（UFL）中，输入是$\mathbb{R}^d$中的一组客户点，目标是通过开设设施并连接客户来最小化总成本（包括设施开设成本与连接成本）。我们研究动态几何流设置，其中客户以网格$\{1,\ldots,\Delta\}^d$中点的插入和删除序列呈现，并聚焦于高维场景——算法必须使用$d\cdot\log\Delta$的多项式空间。我们提出一种基于重要性采样的新算法框架，可在仅使用$\mathrm{poly}(d\cdot\log\Delta)$空间的情况下实现UFL的$O(1)$近似。该框架易于通过两遍扫描实现：第一遍采样点，第二遍估计其贡献。在随机顺序流中，我们通过分别利用流的前后半段，可将此框架扩展为单遍扫描。对于任意顺序流，我们的主要结果通过不同方式结合两遍扫描，实现了单遍$O(d / \log d)$近似。这改进了先前算法（或需$\exp(d)$空间，或仅能保证$O(d\cdot\log^2\Delta)$近似），因此我们的高维算法首次避免了广泛使用的四叉树分解中固有的$O(\log\Delta)$近似因子。这一改进通过采用几何哈希方案实现：该方案将$\mathbb{R}^d$中的点映射到有界直径的桶中，其核心性质是每个直径足够小的点集仅被哈希到少数桶中。通过应用该哈希的另一种界，我们还实现了单遍$O(1 / \epsilon)$近似，空间虽较大但仍为亚线性$O(n^{\epsilon})$（其中$n$为客户数）。最后，我们证明了$1.085$近似需要$\mathrm{poly}(d\cdot\log\Delta)$的指数空间，以此补充结论。