Preprocessing Disks for Convex Hulls, Revisited

2025-02-05

AI Summary

This paper addresses the problem of efficiently reconstructing the convex hull of a set of disks confined to the unit disk. While conventional methods require O(n) reconstruction time, we propose a novel preprocessing framework based on *supersequences*, achieving the first sublinear reconstruction time dependent solely on the number k of unstable disks (where k ≪ n)—optimal O(k) in the worst case. The preprocessing phase constructs the supersequence structure in O(n log n) time, fully decoupling preprocessing from query execution and thereby enhancing efficiency in dynamic settings. We prove the tightness of this time bound and extend the framework to the one-dimensional interval setting, providing complete analogous results. Our core innovation lies in introducing the supersequence as a new auxiliary geometric structure that uniformly captures the influence of both stable and unstable regions, establishing a fresh paradigm for geometric preprocessing.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Topic discovery and tracking

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 $O(nlog n)$ 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.

Problem

• Efficient convex hull reconstruction from preprocessed disks.
• Sublinear time for convex hull using unstable disks.
• Supersequence auxiliary structure decouples preprocessing and reconstruction.

Innovation

• Supersequence as auxiliary structure
• Decoupling preprocessing and reconstruction
• Sublinear time convex hull reconstruction