We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes O(k(\lg{|VP_{\cal P'}(q)|})+(\lg{n'})^{2}+h) (resp. O(k(\lg n')^2+(\lg|VP_{\cal P'}(q)|)+h)) worst-case time to update data structures that store visibility polygon VP_{\cal P'}(q) of a query point q when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain \cal P'. Here, n' is the number of vertices in \cal P', h is the number of obstacles in \cal P', VP_{\cal P'}(q) is the visibility polygon of q in \cal P' (|VP_{\cal P'}(q)| is the number of vertices of VP_{\cal P'}(q)), and k is the number of combinatorial changes in VP_{\cal P'}(q) due to the insertion (resp. deletion) of v. As an application of the above algorithm, we also devise an algorithm for maintaining the visibility graph of a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update data structures that store the visibility graph of the polygonal domain. After preprocessing the initial input polygonal domain, our dynamic algorithm takes O(k(\lg{n'})^{2}+h) (resp. O(k(\lg{n'})^{2}+h)) worst-case time to update data structures that store the visibility graph when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain \cal P'. Here, n' is the number of vertices in \cal P', h is the number of obstacles in \cal P', and k is the number of combinatorial changes in the visibility graph of \cal P' due to the insertion (resp. deletion) of v.

中文翻译：

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平面内动态多边形障碍物之间的可见性多边形和可见性图

我们设计了一种算法来维护动态多边形域中任何查询点的可见性多边形，即当多边形域通过对其障碍物的顶点插入和删除进行修改时，我们更新了存储查询点可见性多边形的数据结构. 在对初始输入多边形域进行预处理以构建一些数据结构后，我们的算法采用 O(k(\lg{|VP_{\cal P'}(q)|})+(\lg{n'})^{2 }+h) (resp. O(k(\lg n')^2+(\lg|VP_{\cal P'}(q)|)+h)) 更新存储可见性的数据结构的最坏情况时间当任何顶点 v 插入（或删除）当前多边形域 \cal P' 的任何障碍物时，查询点 q 的多边形 VP_{\cal P'}(q)。这里，n'是\cal P'中的顶点数，h是\cal P'中的障碍物数量，VP_{\cal P' }(q)是q在\cal P'中的可见多边形（|VP_{\cal P'}(q)|是VP_{\cal P'}(q)的顶点数），k是由于 v 的插入（或删除），VP_{\cal P'}(q) 中组合变化的数量。作为上述算法的应用，我们还设计了一种算法来维护动态多边形域的可见性图，即，当多边形域通过对其障碍物的顶点插入和删除进行修改时，我们更新了存储多边形域可见性图的数据结构。在对初始输入多边形域进行预处理后，我们的动态算法采用 O(k(\lg{n'})^{2}+h) (resp. O(k(\lg{n'})^{2}+h )) 更新存储可见性图的数据结构的最坏情况时间，当任何顶点 v 被插入到（resp. 从当前多边形域\cal P'的任何障碍中删除。这里，n' 是\cal P' 中的顶点数，h 是\cal P' 中障碍物的数量，k 是\cal P' 的可见性图中由于插入（resp . 删除）的 v。