距离敏感平面点定位
Distance-Sensitive Planar Point Location
摘要：
假设$\mathcal{S}$是一个连接的平面多边形细分与$n$边缘，我们要预处理的点位置查询，其中我们给出的概率$γ_i$查询点位于多边形$P_i$的$\mathcal{S}$。我们展示了如何预处理$\mathcal{S}$，使得点~ $p\in P_i$的查询时间取决于~ $γ_i$，此外，还取决于从$p$到~ $P_i$边界的距离-离边界越远，查询速度越快。更准确地说，我们表明，一个点的位置查询可以回答时间$O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{γ_i Δ_{p}^2}\right)\right)$，其中$Δ_{p}$是最短的欧几里得距离的查询点~ $p$的边界$P_i$。我们的结构使用$O(n)$空间和$O(n \log n)$预处理时间。它基于将$\mathcal{S}$的区域分解为凸四边形和三角形，具有以下性质：对于任何点$p\in P_i$，包含$p$的四边形或三角形具有面积$Ω(Δ_{p}^2)$。对于$\mathcal{S}$是单位正方形和$γ_i=\mathrm{area}(P_i)$的细分的特殊情况，我们提出了一个更简单的解决方案，实现了$O\left(\min \left(\log n, \log \frac{1}{Δ_{p}^2}\right)\right)$的查询时间。后一种解决方案可以扩展到三维的凸细分。
英文摘要：
Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $γ_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$γ_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{γ_i Δ_{p}^2}\right)\right)$, where $Δ_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $Ω(Δ_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $γ_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{Δ_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions.
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