Given a set $S$ of $m$ point sites in a simple polygon $P$ of $n$ vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for $S$ in $P$. It is known that the problem has an $\Omega(n+m\log m)$ time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that can solve the problem in $O(n+m\log m)$ expected time. The previous best deterministic algorithms solve the problem in $O(n\log \log n+ m\log m)$ time [Oh, Barba, and Ahn, SoCG 2016] or in $O(n+m\log m+m\log^2 n)$ time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm of $O(n+m\log m)$ time, which is optimal. This answers an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.

中文翻译：

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简单多边形中测地最远点Voronoi图的最佳确定性算法

给定在$ n $顶点的简单多边形$ P $中的$ m $点位置的集合，我们考虑为$ P $中的$ S $计算测地线最远点Voronoi图的问题。已知问题的时间下限为\\ Omega（n + m \ log m）$。以前，有人提出了一种随机算法[Barba，SoCG 2019]，可以在$ O（n + m \ log m）$预期时间内解决问题。先前的最佳确定性算法可解决$ O（n \ log \ log n + m \ log m）$时间[Oh，Barba，and Ahn，SoCG 2016]或$ O（n + m \ log m + m \ log ^ 2 n）$时间[Oh and Ahn，SoCG 2017]。在本文中，我们提出了确定时间为$ O（n + m \ log m）$的确定性算法。这回答了二十年前米切尔在《计算几何手册》中提出的一个开放性问题。