AI 中文总结
本文针对计算几何中凸体高效表示的基础问题，提出基于希尔伯特度量衍生结构的近似成员测试新方法，推导了Delone集规模界，设计了椭球覆盖查询结构，支持指针搜索。
AI 中文摘要
高维空间中凸体的高效表示是计算几何领域的基础问题，近期多项关键进展借助了Macbeath区域相关构造。本文提出一种用于近似成员测试的新型内在方法，整个研究基于与$\boldsymbol{R}^d$中凸体$K$关联的希尔伯特度量衍生结构展开。首先，我们重新审视经济Delone集的构造，基于体积熵概念推导其规模界；其次，设计一种基于椭球简单覆盖的新型查询结构，通过光线投射完成查询响应。额外的优势在于，这种内在视角支持指针搜索，其查询时间可由希尔伯特度量下的移动距离界定。
英文摘要
The efficient representation of convex bodies in multi-dimensional spaces is a fundamental problem in computational geometry. Several key developments were recently brought about using a number of constructions utilizing Macbeath regions. In this paper, we present a novel intrinsic approach for approximate membership testing, where we carry out the entire development based on structures derived from the Hilbert metric associated with a convex body $K$ in $\mathbb{R}^d$. First, we revisit the construction of economical Delone sets, deriving the size bound based on the notion of volume entropy. Second, we design a new query structure based on a simple covering by ellipsoids, where queries are answered by ray shooting. As an added bonus, the intrinsic viewpoint facilitates finger searching, where the query time can be bounded by the distance traveled in the Hilbert metric.
CommentsPresented at SOSA 2024
Journal refSymposium on Simplicity in Algorithms (2024) 286-298
