Chan [JACM, 2010] gave a data structure for maintaining the convex hull of a dynamic set of 3D points under insertions and deletions, supporting extreme-point queries. Subsequent refinements by Kaplan, Mulzer, Roditty, Seiferth, and Sharir [DCG, 2020] and Chan [DCG, 2020] preserved the framework while improving bounds. The current best result achieves $O(\log^2 n)$ amortized insertion time, $O(\log^4 n)$ amortized deletion time, and $O(\log^2 n)$ worst-case query time. These techniques also yield dynamic 2D Euclidean nearest neighbor searching via duality, where the problem becomes maintaining the lower envelope of 3D planes for vertical ray-shooting queries. Using randomized vertical shallow cuttings, Kaplan et al. [DCG, 2020] and Liu [SICOMP, 2022] extended the framework to dynamic lower envelopes of general 3D surfaces, obtaining the same asymptotic bounds and enabling nearest neighbor searching under general distance functions. We revisit Chan's framework and present a modified structure that reduces the deletion time to $O(\log^3 n \log\log n)$, while retaining $O(\log^2 n)$ amortized insertion time, at the cost of increasing the query time to $O(\log^3 n / \log\log n)$. When the overall complexity is dominated by deletions, this yields an improvement of roughly a logarithmic factor over previous results.

翻译：Chan [JACM, 2010] 提出了一种数据结构，用于在插入和删除操作下维护动态三维点集的凸包，并支持极值点查询。后续由 Kaplan、Mulzer、Roditty、Seiferth 和 Sharir [DCG, 2020] 以及 Chan [DCG, 2020] 进行的改进保留了原有框架，同时提升了性能界限。当前最佳结果实现了 $O(\log^2 n)$ 的平摊插入时间、$O(\log^4 n)$ 的平摊删除时间以及 $O(\log^2 n)$ 的最坏情况查询时间。这些技术也通过对偶性实现了动态二维欧几里得最近邻搜索，其中问题转化为维护三维平面的下包络以支持垂直射线射击查询。利用随机化垂直浅切割，Kaplan 等人 [DCG, 2020] 和 Liu [SICOMP, 2022] 将该框架扩展至一般三维曲面的动态下包络维护，获得了相同的渐进复杂度界限，并使得在一般距离函数下的最近邻搜索成为可能。我们重新审视 Chan 的框架，提出一种改进结构，将删除时间降低至 $O(\log^3 n \log\log n)$，同时保持 $O(\log^2 n)$ 的平摊插入时间，代价是将查询时间增加至 $O(\log^3 n / \log\log n)$。当整体复杂度主要由删除操作主导时，相较于先前结果，这带来了大约一个对数因子的改进。