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.

翻译：在预处理框架中，给定一组允许进行预处理的区域，通过构建辅助结构，使得当这些区域各给出一个实现点（每个区域一个点）时，该辅助结构能够用于重建所需的几何结构，且效率高于未经预处理的情况。先前的研究表明，一组具有常数层叠度的 $n$ 个单位圆盘可在 $O(n\log n)$ 时间内完成预处理，从而使得任意实现的凸包可在 $O(n)$ 时间内重建。（该研究主要关注三角剖分，凸包是其副产品。）本工作中，我们首次证明凸包的重建时间可与\emph{不稳定}圆盘的数量成正比，该数量可能为亚线性，且此运行时间是最优的。此处，若凸包的组合结构不依赖于其实现点的位置，则称该圆盘为\emph{稳定}的。我们实现该结果的主要工具是在预处理阶段构建超序列作为辅助结构，即输出圆盘的一个超序列，使得任意实现的凸包为其子序列。使用超序列作为辅助结构的一个优势是，它允许我们在比先前工作更强的意义上将预处理阶段与重建阶段解耦，从而形成两个独立的算法问题，各自可能具有独立的研究意义。最后，在为凸包问题获得结果的过程中，我们解决了在一维区间上构建此类超序列的对应问题，并给出了该情况下的相应结果。