优先权矩阵和矩形生长的快速算法
Faster Algorithms for Growing Prioritized Disks and Rectangles
摘要：
受地图标注的启发，Funke、Krumpe和Storandt [IWOCA 2016]引入了以下问题：我们在平面上给定了一系列$n$圆盘。最初，所有磁盘的半径都是$0$，它们以恒定但可能不同的速度增长。每当两个磁盘接触时，具有较高索引的磁盘就会消失。目标是确定排除顺序，即，磁盘消失的顺序。我们为这个问题提供了第一个一般次二次算法。我们的解决方案扩展到其他形状（例如，矩形），它可以在任何固定的维度上工作。我们还描述了一种替代算法，是基于四叉树。它的运行时间是$O\big(n \big(\log n + \min \{ \log Δ, \log Φ\}\big)\big)$，其中$Δ$是最快和最慢的增长率之比，$Φ$是两个磁盘中心之间的最大和最小距离之比。这改善了Funke、Krumpe和Storandt [IWOCA 2016]、Bahrdt等人[ALENEX 2017]以及Funke和Storandt [EuroCG 2017]先前算法的运行时间。最后，我们给出了一个$Ω(n\log n)$下界，表明我们的四叉树算法几乎是紧的。
英文摘要：
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 $O\big(n \big(\log n + \min \{ \log Δ, \log Φ\}\big)\big)$, where $Δ$ is the ratio of the fastest and the slowest growth rate and $Φ$ 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 $Ω(n\log n)$ lower bound, showing that our quadtree algorithms are almost tight.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
