# Faster Algorithms for Growing Prioritized Disks and Rectangles

## TL;DR

This paper introduces the first subquadratic algorithm for the growing prioritized disks problem, extending to other shapes and dimensions, with improved running times and near-tight bounds, motivated by map labeling applications.

## Contribution

It presents the first general subquadratic algorithm for growing prioritized disks, extends the approach to other shapes and dimensions, and provides a quadtree-based algorithm with improved efficiency.

## Key findings

- First subquadratic algorithm for the problem
- Extension to rectangles and higher dimensions
- Quadtree algorithm with near-optimal bounds

## Abstract

Motivated by map labeling, Funke, Krumpe, and Storandt [IWOCA 2016] introduced the following problem: we are given a sequence of $n$ disks in the plane. Initially, all disks have radius $0$, and they grow at constant, but possibly different, speeds. Whenever two disks touch, the one with the higher index disappears. The goal is to determine the elimination order, i.e., the order in which the disks disappear. We provide the first general subquadratic algorithm for this problem. Our solution extends to other shapes (e.g., rectangles), and it works in any fixed dimension. We also describe an alternative algorithm that is based on quadtrees. Its running time is $O\big(n \big(\log n + \min \{ \log \Delta, \log \Phi \}\big)\big)$, where $\Delta$ is the ratio of the fastest and the slowest growth rate and $\Phi$ is the ratio of the largest and the smallest distance between two disk centers. This improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EuroCG 2017]. Finally, we give an $\Omega(n\log n)$ lower bound, showing that our quadtree algorithms are almost tight.