简单多边形中近似最远邻查询的核集
A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon
AI总结：
本研究针对简单多边形内的近似最远邻查询问题，证明存在大小为O(1/ε²)的核集，可支持存储需求与点集规模无关的ε-近似查询，并给出了相应的核集构建时间复杂度。
摘要：
设$\u27fbP$为具有$m$个顶点的简单多边形，$P$为$\u27fbP$内部的$n$个点构成的集合。我们证明，对任意$\u03b5>0$，存在大小为$O(1/\u03b5^2)$的集合$\u27fbC \u2282 P$，满足：对多边形$\u27fbP$内的任意查询点$q$，$q$到其在$\u27fbC$中最远邻的测地距离，至少为到其在$P$中最远邻的测地距离的$1-\u03b5$倍。因此，集合$\u27fbC$可用于回答$\u03b5$-近似最远邻查询，对应数据结构的存储需求与$P$的大小无关。该核集可在$O\left(\frac{1}{\u03b5} \left( n\log(1/\u03b5) + (n+m)\log(n+m)\right) \right)$时间内构建。
英文摘要：
Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$ times the geodesic distance to its further neighbor in $P$. Thus the set $\mathcal{C}$ can be used for answering $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of $P$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
