Instance-Optimal Imprecise Convex Hull

2025-04-03

AI Summary

This paper studies the convex hull reconstruction problem for imprecise point sets in the plane, where each point is represented by a region; the objective is to minimize the number of region queries while efficiently determining the cyclic order of convex hull vertices. We propose the first algorithm that simultaneously achieves query optimality and time efficiency: it performs exactly $r(F,P)$ queries—matching the instance-specific lower bound—and runs in $O(r log^3 n)$ time. Our approach integrates geometric divide-and-conquer, interval trees, and dynamic convex hull maintenance, supporting general region models including polygons, simple $k$-gons, and disjoint disks with bounded radii. Crucially, even under arbitrary region overlap, the algorithm retains $O(r log^3 n)$ runtime and $O(r)$ query complexity. Compared to prior work, per-query processing time improves from polynomial to quasi-logarithmic, and the total number of queries significantly outperforms existing near-linear-time algorithms.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Imprecise measurements of a point set P = (p1, ..., pn) can be modelled by a family of regions F = (R1, ..., Rn), where each imprecise region Ri contains a unique point pi. A retrieval models an accurate measurement by replacing an imprecise region Ri with its corresponding point pi. We construct the convex hull of an imprecise point set in the plane, where regions in F may be retrieved at unit cost. The goal is to determine the cyclic ordering of the convex hull vertices of P as efficiently as possible. Here, efficiency is interpreted in two ways: (i) minimising the number of retrievals, and (ii) computing each retrieval location quickly. Prior works focused on only one of these two aspects: either minimising retrievals or optimising algorithmic runtime. Our contribution is the first to simultaneously achieve both. Let r(F, P) denote the minimal number of retrievals required by any algorithm to determine the convex hull of P for a given instance (F, P). For a family F of n constant-complexity polygons, our main result is a reconstruction algorithm that performs O(r(F, P)) retrievals in O(r(F, P) log^3 n) time. Compared to previous approaches that achieve optimal retrieval counts, we improve the runtime per retrieval by a exponential factor, from polynomial to polylogarithmic. Compared to near-linear time algorithms, we significantly reduce the number of retrievals used, and broaden the input families to include overlapping regions. We further extend our results to simple k-gons and to pairwise disjoint disks with radii in [1,k], where our runtime scales linearly with k.

Problem

• Construct convex hull from imprecise point sets efficiently
• Minimize retrievals and optimize runtime simultaneously
• Handle overlapping regions and extend to complex polygons

Innovation

• Minimizes retrievals and optimizes runtime simultaneously
• Achieves polylogarithmic time per retrieval
• Extends to overlapping regions and various shapes