We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+\delta})$ for an arbitrarily small constant $\delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(n\log n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(n\log n(\log\log n)^{O(1)})$ result [SoCG 2020].
翻译:我们研究的几何数据集覆盖动态设置中的问题,允许插入和删除点和对象。我们展示了第一个动态数据结构,可以在亚线性更新时保持2D中轴对齐正方形设定覆盖值的O(1)美元准比。更准确地说,我们获得任意的小常数$\delta>0的随机更新时间$O(n ⁇ 2/3 ⁇ *delta})。以前,Agarwal、Chang、Suri、Xiao和Sue[SoCG 2020]所知道的具有亚线性更新时间的动态数据集覆盖单位方形的亚线性数据结构。如果只需要解决方案的大致大小,那么我们也可以获得2D和3D中半空格磁盘的亚线性更新时间。作为副产品,我们的动态集技术覆盖还产生一个最佳的随机化$O(n\log n)$-时间算法,用于2D磁盘和3D半空格的静态覆盖值,改进了我们早期的$O(nlog n\log\ nG)\=O}[结果。