Computer Science > Computational Geometry
[Submitted on 5 Nov 2020 (v1), last revised 5 Apr 2023 (this version, v3)]
Title:Visibility Extension via Reflection
View PDFAbstract: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 $\pi$. 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 { $\bf \lceil \frac{\alpha}{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $\alpha$ 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.
Submission history
From: Arash Vaezi [view email][v1] Thu, 5 Nov 2020 21:42:03 UTC (741 KB)
[v2] Thu, 2 Dec 2021 13:16:51 UTC (3,037 KB)
[v3] Wed, 5 Apr 2023 12:09:49 UTC (1,342 KB)
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