重新审视凸包的磁盘预处理
Preprocessing Disks for Convex Hulls, Revisited
AI总结：
本研究重新审视凸包的磁盘预处理问题，提出通过超序列辅助结构实现更高效的凸包重建，且运行时间与不稳定磁盘数量成比例。
摘要：
在预处理框架中，给定一组区域，允许对其进行预处理以创建某种辅助结构，使得当这些区域的实现给出一个点时，该辅助结构可以更高效地重建所需的几何结构。先前的工作表明，一组n个常数ply的单位圆可以以O(n log n)的时间预处理，使得任何实现的凸包可以以O(n)的时间重建。（先前的工作侧重于三角剖分，凸包是附带结果。）在本工作中，我们首次展示了可以以与不稳定磁盘数量成比例的时间重建凸包，该时间可能为次线性，并且这种运行时间是最佳的。这里，如果凸包的组合结构不依赖于其实现点的位置，则称为稳定磁盘。我们实现结果的主要工具是使用超序列作为预处理阶段构建的辅助结构，即我们输出磁盘的超序列，使得任何实现的凸包是该序列的子序列。使用超序列作为辅助结构的优点是它允许我们在更强的意义上将预处理阶段与重建阶段解耦，从而得到两个可能具有独立兴趣的算法问题。最后，在获得凸包结果的过程中，我们解决了相应的一维区间创建此类超序列的问题，从而得到相应结果。
英文摘要：
In the preprocessing framework one is given a set of regions that one is allowed to preprocess to create some auxiliary structure such that when a realization of these regions is given, consisting of one point per region, this auxiliary structure can be used to reconstruct some desired output geometric structure more efficiently than would have been possible without preprocessing. Prior work showed that a set of $n$ unit disks of constant ply can be preprocessed in $O(n\log n)$ time such that the convex hull of any realization can be reconstructed in $O(n)$ time. (This prior work focused on triangulations and the convex hull was a byproduct.) In this work we show for the first time that we can reconstruct the convex hull in time proportional to the number of \emph{unstable} disks, which may be sublinear, and that such a running time is the best possible. Here a disk is called \emph{stable} if the combinatorial structure of the convex hull does not depend on the location of its realized point. The main tool by which we achieve our results is by using a supersequence as the auxiliary structure constructed in the preprocessing phase, that is we output a supersequence of the disks such that the convex hull of any realization is a subsequence. One advantage of using a supersequence as the auxiliary structure is that it allows us to decouple the preprocessing phase from the reconstruction phase in a stronger sense than was possible in previous work, resulting in two separate algorithmic problems which may be independent interest. Finally, in the process of obtaining our results for convex hulls, we solve the corresponding problem of creating such supersequences for intervals in one dimension, yielding corresponding results for that case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
