Faster algorithms for growing prioritized disks and rectangles

Hee Kap Ahn, Sang Won Bae, Jongmin Choi, Matias Korman, Wolfgang Mulzer, Eunjin Oh, Ji won Park, André van Renssen, Antoine Vigneron

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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(n(log⁡n+min⁡{log⁡Δ,log⁡Φ})), 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(nlog⁡n) lower bound, showing that our quadtree algorithms are almost tight.

Original languageEnglish
Pages (from-to)23-39
Number of pages17
JournalComputational Geometry: Theory and Applications
Volume80
DOIs
Publication statusPublished - 2019 Jul

Keywords

  • Condition number
  • Data structures
  • Elimination order
  • Lower bound
  • Quadtree

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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