This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by k real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We present an efficient $O\left(k^2\right)$ time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM n points set, we present an $O(\log nk+mk)$ time query and $O\left(nk\right)$ space algorithm, where m is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a $O(k^2+\raisebox{1ex}{$(k\log n\log \log n)$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.+m)$ time and $O(\raisebox{1ex}{$n^2$}\!\left/ \!\raisebox{-1ex}{$\log n$}\right.+\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.)$ space approximation algorithm, where $0<\epsilon \le 1$ is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two $O(k^2+\raisebox{1ex}{$(k\log nk\log \log nk)$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.+mk)$ and $O(m{k}^2+\raisebox{1ex}{$(k\log nk\log \log nk)$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.)$ time query and $O(\raisebox{1ex}{${\left(nk\right)}^2$}\!\left/ \!\raisebox{-1ex}{$\log nk$}\right.+\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.)$ space approximation algorithms for the independent and dependent case, respectively.

本文解决了存在几何不确定性的情况下平面中点的一系列几何半平面检索查询。这些问题包括精确和不确定的点集以及由精确或不确定的线定义的半平面查询，其位置不确定性是独立的或相关的，并且由k 个实值参数定义。点坐标不确定性使用线性参数几何不确定性模型 (LPGUM) 进行建模，该模型是一种富有表现力且计算效率高的最坏情况、几何不确定性的一阶线性近似，支持点位置之间的参数依赖性。我们提出了一种高效的$氧（k^2）$用于计算定义半平面查询的 LPGUM 线包络线的时间和空间算法。对于一条精确的直线和一个 LPGUM n点集，我们提出一个$氧（\mathrm{日志}nk+米k）$时间查询和$氧（nk）$空间算法，其中m是半平面线上或上方的 LPGUM 点的数量。对于 LPGUM 线和精确点集，我们提出$氧（k^2+\raisebox{1ex}{$（k\mathrm{日志}n\mathrm{日志}\mathrm{日志}n）$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.+米）$时间和$氧（\raisebox{1ex}{$n^2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{日志}n$}\right.+\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.）$空间近似算法，其中$0<\epsilon \le 1$是期望的近似误差。对于 LPGUM 线和 LPGUM 点集，我们提出两个$氧（k^2+\raisebox{1ex}{$（k\mathrm{日志}nk\mathrm{日志}\mathrm{日志}nk）$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.+米k）$和$氧（米k^2+\raisebox{1ex}{$（k\mathrm{日志}nk\mathrm{日志}\mathrm{日志}nk）$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.）$时间查询和$氧（\raisebox{1ex}{${（nk）}^2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{日志}nk$}\right.+\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.）$分别针对独立和非独立情况的空间近似算法。