完美守卫海峡：弱可见多边形精确算法

中文摘要 AI 辅助

艺术画廊问题（AGP）要求寻找最少数量的守卫，使得它们能看到一个简单多边形的所有部分。该问题是$\exists\mathbb{R}$-完全的，因此是NP困难的。我们证明，对于一类特定的多边形，将守卫限制在一条边上可以使AGP被精确且高效地求解。我们称此为海峡守卫问题（SGP）。其输入是一个弱可见多边形（WVP）：一个简单多边形，其中每个点都能从某条固定边（称为底边）上的某个点看到。SGP在底边上放置最少数量的守卫，使得它们共同看到整个多边形。首先，一个结构事实：底边上覆盖边界的守卫已经覆盖了整个内部，从而将二维覆盖问题转化为一维问题。我们的主要结果是见证守卫算法，该算法在$O((n + \mathrm{OPT} \cdot \rho)(\log n + \log \mathrm{OPT}))$时间内精确求解SGP，其中$\rho$是WVP中反射顶点的数量，OPT是最小守卫数量。该算法是输出敏感的，并通过从其输出中导出的规模为OPT的见证集来证明最优性。我们还研究了守卫顶点版本，并证明了紧的$\Theta(n \log n)$界，其中下界由排序归约得出。作为SGP的推论，我们获得了关于海拔地形守卫（ATG）的两个结果，而SGP是ATG的推广。我们给出了一个线性时间的完美守卫算法，改进了Daescu、Friedrichs、Malik、Polishchuk和Schmidt之前的$O(n^2 \log n)$界。我们还解决了他们关于最小守卫海拔的问题，在$O(nk + k^2 \log k)$时间内完成，改进了Kang、Kim和Ahn的$O(k^2 \lambda_{k-1}(n) \log n)$界。

英文摘要

The Art Gallery Problem (AGP) asks for the fewest guards that see all of a simple polygon. It is $\exists\mathbb{R}$-complete, hence NP-hard. We show that for a particular class of polygons, confining guards to a single edge makes AGP exactly and efficiently solvable. We call this the Strait Guarding Problem (SGP). Its input is a weak visibility polygon (WVP): a simple polygon where every point is seen from some point of one fixed edge, the base. SGP places the fewest guards on the base that jointly see the whole polygon. First, a structural fact: guards on the base edge that cover the boundary already cover the entire interior, turning a two-dimensional covering problem into a one-dimensional one. Our main result is the Witness-Guard Algorithm, which solves SGP exactly in $O((n + \mathrm{OPT} \cdot ρ)(\log n + \log \mathrm{OPT}))$ time, where $ρ$ is the number of reflex vertices in the WVP and OPT is the minimum number of guards. It is output-sensitive and certifies optimality by a witness set of size OPT derived from its output. We also study the guarding-the-vertex version and prove a tight $Θ(n \log n)$ bound, with the lower bound following from Sorting. As a corollary of SGP, we obtain two results for altitude terrain guarding (ATG), a special case that SGP generalizes. We give a linear-time perfect-guarding algorithm, improving the previous $O(n^2 \log n)$ bound of Daescu, Friedrichs, Malik, Polishchuk and Schmidt. We also resolve their problem on the minimum guarding altitude, in $O(nk + k^2 \log k)$ time, improving on the $O(k^2 λ_{k-1}(n) \log n)$ bound of Kang, Kim and Ahn.

发表机构

• Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India（印度统计学院先进计算与微电子单元）

机构由 AI 辅助整理，请以论文原文为准。