论文标题
通过反射扩展可见性
Visibility Extension via Reflection
论文作者
论文摘要
本文研究了美术馆问题的变体，其中``墙壁''可以用\ emph {反射边缘}代替，这使守卫可以进一步看到，从而看到了较大的画廊。给定一个简单的polygon $ \ cal p $，首先，我们将一个守卫视为一名守护者，我们将一定的vive vise vive添加到了一定的范围内，我们将其视为vive vive vive的一个vive，以添加一定的vive views，以供您使用。镜面反射是镜像的镜面和弥漫性反射，在反射的弥漫类型中，事件和反射射线之间的角度可能会假设所有可能的值$ 0 $ $ 0 $。 $ np $ -hard也是如此。 其次，我们假设所有边缘都是反射器，我们打算减少覆盖整个画廊所需的最小警卫数量。 Chao Xu证明，即使考虑到$ r $ oppecular的反射，也可能需要$ \ lfloor \ frac {n} {3} {3} \ rfloor $ guards来覆盖多边形。令$ r $为警卫可见度射线的最大反射数。 在这项工作中，我们证明，考虑$ r $ \ emph {fiffuse}反思，覆盖给定的简单的多边形$ \ cal p $所需的最小数量\ emph {dertex或界限}守卫减少到{$ \ bf \ bf \ lceil \ lceil \fracα{1+ \ lfloor \ lfloor \ frlo \ frac} \ rceil $}，其中$α$表示无需反射而覆盖多边形所需的最小警卫数。我们还概括了$ \ mathcal {o}（\ log n）$ - 在存在反射的情况下，顶点保护问题的近似值算法可以工作。
This paper studies a variant of the Art Gallery problem in which the ``walls" can be replaced by \emph{reflecting edges}, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon $\cal P$, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between $0$ and $π$. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are $NP$-hard. We show that several cases of adding an area to the visible area of a given point guard are $NP$-hard, too. Second, we assume all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery. Chao Xu proved that even considering $r$ specular reflections, one may need $\lfloor \frac{n}{3} \rfloor$ guards to cover the polygon. Let $r$ be the maximum number of reflections of a guard's visibility ray. In this work, we prove that considering $r$ \emph{diffuse} reflections, the minimum number of \emph{vertex or boundary} guards required to cover a given simple polygon $\cal P$ decreases to { $\bf \lceil \fracα{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $α$ indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.
