Shallow cuttings are a fundamental tool in computational geometry and spatial databases for solving offline and online range searching problems. For a set $P$ of $N$ points in 3-D, at SODA'14, Afshani and Tsakalidis designed an optimal $O(N\log_2N)$ time algorithm that constructs shallow cuttings for 3-D dominance ranges in internal memory. Even though shallow cuttings are used in the I/O-model to design space and query efficient range searching data structures, an efficient construction of them is not known till now. In this paper, we design an optimal-cost algorithm to construct shallow cuttings for 3-D dominance ranges. The number of I/Os performed by the algorithm is $O\left(\frac{N}{B}\log_{M/B}\left(\frac{N}{B}\right) \right)$, where $B$ is the block size and $M$ is the memory size. As two applications of the optimal-cost construction algorithm, we design fast algorithms for offline 3-D dominance reporting and offline 3-D approximate dominance counting. We believe that our algorithm will find further applications in offline 3-D range searching problems and in improving construction cost of data structures for 3-D range searching problems.

翻译：浅层剖分是计算几何与空间数据库中解决离线与在线范围搜索问题的基本工具。针对三维空间中的点集$P$（含$N$个点），Afshani与Tsakalidis在SODA'14上提出了内部存储器中面向三维支配域浅层剖分的最优$O(N\log_2N)$时间算法。尽管浅层剖分已被用于I/O模型中以设计空间高效且查询高效的范围内存数据结构，但目前尚无高效的构造方法。本文设计了面向三维支配域浅层剖分的最优代价构造算法。该算法执行的I/O次数为$O\left(\frac{N}{B}\log_{M/B}\left(\frac{N}{B}\right) \right)$，其中$B$为块大小，$M$为内存容量。作为该最优代价构造算法的两项应用，我们设计了面向离线三维支配域报告与离线三维近似支配域计数的快速算法。我们相信，该算法将在离线三维范围搜索问题及改善三维范围搜索数据结构构造代价方面获得进一步应用。