Preprocessing Disks for Convex Hulls, Revisited

TL;DR

This work studies the preprocessing framework for geometric uncertainty, focusing on rebuilding the convex hull of a realization of unit disks with bounded ply after a preprocessing phase. The authors introduce the concept of a supersequence as a versatile auxiliary structure that decouples preprocessing from reconstruction and enable reconstruction time that scales with the number of unstable disks, including sublinear cases; they also partition the 2D hull problem into four quarter-hulls and solve a 1D interval sorting subproblem to support the 2D results. They establish existence and construction results for smooth quarter-hull-supersequences of size $O(\Delta^2 n)$ and show reconstruction in $O(\alpha\beta|\Xi|)$ time, with sublinear reconstruction in $O(\alpha\beta\mu)$ when only $\mu$ disks are unstable, providing matching lower-bound intuition. The paper also treats disjoint and overlapping unit disks, develops a detailed 1D preprocessing pipeline for intervals, and discusses limitations and open problems, including removing the smoothness requirement and extending to more general region families. Overall, the approach delivers a clean, modular framework that ties 2D geometric reconstruction to a robust 1D subroutine, offering practical sublinear-time possibilities under uncertainty and paving the way for broader applications beyond convex hulls.

Abstract

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.

Figures (11)

• Figure 1: A set of disjoint unit disks $\mathcal{R}\xspace$ can be preprocessed into an auxiliery structure $\Xi(\mathcal{R}\xspace)$ in $O(n\log n)$ time, such that the convex hull of a set of points $P$ that respects $\mathcal{R}\xspace$ can be computed in linear time using $\Xi(\mathcal{R}\xspace)$ (compared to $\Theta(n \log n)$ time without preprocessing) held2008triangulatingvan2010preprocessingBLMM11.
• Figure 2: (a) A set of disjoint unit disks $\mathcal{R}\xspace$. (b) All possible combinatorial convex hulls. We identify five types of disks: (i) stable impossible interior disks that never contribute to the convex hull (red); (ii) unstable potential interior disks that may or may not contribute (orange); (ii) unstable potential boundary disks that may or may not contribute (yellow); (iii) unstable guaranteed boundary disks that are guaranteed to contribute, but for which the location may influence the structure of the hull (blue); and (iv) stable guaranteed boundary disks that are guaranteed to contribute, and the structure is independent of their location (green). (Refer to Section \ref{['sec:classifications']} for precise definitions).
• Figure 3: (a) A set of disjoint unit disks $\mathcal{R}\xspace$ and (b) a sequence of disks that is guaranteed to contain the vertices of the convex hull in the correct order. (c) A possible true convex hull.
• Figure 4: The same example as Figure \ref{['fig:intro-types']} with the regions $I$ and $E$ drawn in.
• Figure 5: An example of a sequence of unit intervals of ply $\Delta=3$.

Theorems & Definitions (40)

• Definition 1
• Definition 3
• Definition 4
• Lemma 5
• Lemma 6
• Definition 7
• Definition 8
• Theorem 9
• Theorem 10
• Theorem 11