Given an orthogonal polygon with orthogonal holes, we devise a dynamic algorithm for guarding with vertex guards, i.e., whenever orthogonal polygon is modified, algorithm updates the set of vertex guards and their positions for guarding the modified orthogonal polygon. Our algorithm modifies the guard placement locally while ensuring the updated orthogonal polygon with $h$ holes and $n$ vertices, is guarded using at most $\lfloor (n+2h)/4 \rfloor$ vertex guards. The algorithm to update vertex guards after any modification to the polygon takes $O(k \lg{(n+ n')})$ amortized time. Here, $n'$ and $n$ are the number of vertices of the orthogonal polygon before and after the update, respectively; and, $k$ is the sum of the number of vertices added to or removed from the orthogonal polygon, the number of cuts in the L-shaped partitioning of the free space of the orthogonal polygon that got affected due to the update, and the number of channels affected due to the update. For the special case of the initial orthogonal polygon being hole-free, and each update resulting in a hole-free orthogonal polygon, our dynamic guard updating algorithm takes $O(k\lg{(n+n')})$ worst-case time. Initially, we preprocess the input orthogonal polygon with $q$ vertices in $O(q \lg{q})$ time to construct data structures of size $O(q\frac{\lg{q}}{\lg\lg{q}})$.

中文翻译：

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动态正交艺术画廊的顶点保护

给定一个带有正交孔的正交多边形，我们设计了一种动态算法来保护顶点保护，即，无论何时修改正交多边形，算法都会更新顶点保护集及其位置以保护修改后的正交多边形。我们的算法在本地修改保护位置，同时确保更新的带有 $h$ 孔和 $n$ 顶点的正交多边形受到最多 $\lfloor (n+2h)/4 \rfloor$ 顶点保护。在对多边形进行任何修改后更新顶点保护的算法需要 $O(k \lg{(n+ n')})$ 摊销时间。其中，$n'$和$n$分别为更新前后正交多边形的顶点数；并且，$k$ 是添加到正交多边形或从正交多边形中删除的顶点数的总和，由于更新而受到影响的正交多边形的自由空间的 L 形分区中的切割数，以及因更新而受到影响的通道数。对于初始正交多边形无孔的特殊情况，并且每次更新都会导致无孔正交多边形，我们的动态保护更新算法采用 $O(k\lg{(n+n')})$ 最坏情况-案件时间。最初，我们在 $O(q \lg{q})$ 时间内对输入的带有 $q$ 个顶点的正交多边形进行预处理，以构建大小为 $O(q\frac{\lg{q}}{\lg\lg {q}})$。我们的动态守卫更新算法需要 $O(k\lg{(n+n')})$ 最坏情况时间。最初，我们在 $O(q \lg{q})$ 时间内对输入的带有 $q$ 个顶点的正交多边形进行预处理，以构建大小为 $O(q\frac{\lg{q}}{\lg\lg {q}})$。我们的动态守卫更新算法需要 $O(k\lg{(n+n')})$ 最坏情况时间。最初，我们在 $O(q \lg{q})$ 时间内对输入的带有 $q$ 个顶点的正交多边形进行预处理，以构建大小为 $O(q\frac{\lg{q}}{\lg\lg {q}})$。