Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition

TL;DR

This paper introduces a data structure that efficiently computes the onion decomposition of point sets chosen from disjoint disks, enabling fast layered convex hull queries with optimal bounds.

Contribution

It presents a novel linear-size data structure with $O(n ext{log} n)$ preprocessing and $O(n ext{log} k)$ query time for onion decomposition of imprecise points.

Findings

Data structure built in $O(n ext{log} n)$ time

Query time is $O(n ext{log} k)$, optimal with a matching lower bound

Applicable to sets of points with one point per disk in the plane

Abstract

Let $\mathcal{D}$ be a set of $n$ pairwise disjoint unit disks in the plane. We describe how to build a data structure for $\mathcal{D}$ so that for any point set $P$ containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of $P$. Our data structure can be built in $O(n \log n)$ time and has linear size. Given $P$, we can find its onion decomposition in $O(n \log k)$ time, where $k$ is the number of layers. We also provide a matching lower bound. Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onion