AI 中文总结
本研究针对高维ℓ_∞和ℓ_2空间的概率度量分解采样问题，分别提出接近线性时间的填充分解与分离分解算法，提升了同时间复杂度下的性能，相关技术可应用于生成树和最近邻搜索。
AI 中文摘要
度量分解是设计涉及距离的算法的基础工具。我们研究了在高维d的ℓ_∞和ℓ_2空间中，对n个点的集合的概率度量分解进行采样的快速算法。针对ℓ_∞空间，我们设计了一种填充分解算法，运行时间为Õ(nd²)，关于n接近线性，且实现了Õ(log n)的填充参数。该算法构建了一种基于ℓ_∞几何性质的新型稀疏邻域覆盖[Indyk, JCSS'01]，并利用了近期覆盖与分解之间的归约方法[Conroy and Filtser, STOC'25]。针对ℓ_2空间，我们设计了一种分离分解算法，在n^{1+o(1)}的近线性时间内实现了接近最优的Õ(√log n)分离度。我们的结果相比运行时间相近的已知算法提升了Ω(√log n)倍，相关技术还可应用于生成树（spanners）和最近邻搜索领域。
英文摘要
Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of $n$-point sets in $\ell_\infty$ and $\ell_2$ spaces of high dimension $d$. For $\ell_\infty$, we design a padded-decomposition algorithm that runs in time $\tilde{O}(nd^2)$, which is near-linear in $n$, and achieves padding parameter $\tilde{O}(\log n)$. Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of $\ell_\infty$ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For $\ell_2$, we design a separating-decomposition algorithm that achieves near optimal separation $\tilde{O}(\sqrt{\log n})$ in almost-linear time $n^{1+o(1)}$. Our bounds improve over known algorithms with similar running time by a factor $Ω(\sqrt{\log n})$, and the techniques have additional applications to spanners and nearest-neighbor search.
Comments17 pages
