轨迹范围可见性
Trajectory Range Visibility
摘要：
考虑两个具有恒定但不一定相等速度的实体，在简单多边形$P$内的两个给定分段线性轨迹上移动。轨迹范围可见性问题涉及确定两个实体彼此可见的子轨迹。这个问题的一个更直接的决策版本被称为轨迹可见性，其中轨迹是线段。决策版本指定实体是否可以看到彼此。P. Eades等人在2020年研究了这个版本，他们假设实体的速度是恒定的。然而，本文提出的方法支持非恒定复杂性轨迹。此外，我们报告了实体可以看到对方的每一对恒定速度。特别地，对于移动实体的每个恒定速度，我们指定：$(1)$其他实体轨迹的所有可见部分。$(2)$其他实体的所有可能的恒定速度变得可见。关于线段轨迹，我们提出了$\mathcal{O}(n \log n)$运行时间算法，该算法获得所有对子轨迹上的移动实体变得彼此可见，其中$n$是$P$的复杂性。关于一般情况下，我们提供了一个算法与$\mathcal{O}(n \log n + m(\log m + \log n))$运行时间，其中$m$表示两个轨迹的复杂性。我们提供$\mathcal{O}(\log n)$查询时间的线段轨迹和$\mathcal{O}(\log m + k)$的非常数复杂性的s. t。$k$是输出中报告的速度范围数。有趣的是，我们的结果只需要$\mathcal{O}(n + m)$空间的非常数复杂性轨迹。
英文摘要：
Consider two entities with constant but not necessarily equal velocities, moving on two given piece-wise linear trajectories inside a simple polygon $P$. The Trajectory Range Visibility problem deals with determining the sub-trajectories on which two entities become visible to each other. A more straightforward decision version of this problem is called Trajectory Visibility, where the trajectories are line segments. The decision version specifies whether the entities can see one another. This version was studied by P. Eades et al. in 2020, where they supposed given constant velocities for the entities. However, the approach presented in this paper supports non-constant complexity trajectories. Furthermore, we report every pair of constant velocities with which the entities can see each other. In particular, for every constant velocity of a moving entity, we specify: $(1)$ All visible parts of the other entity's trajectory. $(2)$ All possible constant velocities of the other entity to become visible. Regarding line-segment trajectories, we present $\mathcal{O}(n \log n)$ running time algorithm which obtains all pairs of sub-trajectories on which the moving entities become visible to one another, where $n$ is the complexity of $P$. Regarding the general case, we provide an algorithm with $\mathcal{O}(n \log n + m(\log m + \log n))$ running time, where $m$ indicates the complexity of both trajectories. We offer $\mathcal{O}(\log n)$ query time for line segment trajectories and $\mathcal{O}(\log m + k)$ for the non-constant complexity ones s.t. $k$ is the number of velocity ranges reported in the output. Interestingly, our results require only $\mathcal{O}(n + m)$ space for non-constant complexity trajectories.
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