License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.24
URN: urn:nbn:de:0030-drops-138231
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13823/
Go to the corresponding LIPIcs Volume Portal

Chan, Timothy M.

Faster Algorithms for Largest Empty Rectangles and Boxes

pdf-format:
LIPIcs-SoCG-2021-24.pdf (0.8 MB)

Abstract

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time O(nlog²n) for d = 2 (by Aggarwal and Suri [SoCG'87]) and near n^d for d ≥ 3. We describe faster algorithms with running time
- O(n2^{O(log^*n)}log n) for d = 2,
- O(n^{2.5+o(1)}) time for d = 3, and
- Õ(n^{(5d+2)/6}) time for any constant d ≥ 4.
To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee’s measure problem to optimize certain objective functions over the complement of a union of orthants.

BibTeX - Entry

@InProceedings{chan:LIPIcs.SoCG.2021.24,
  author =	{Chan, Timothy M.},
  title =	{{Faster Algorithms for Largest Empty Rectangles and Boxes}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13823},
  URN =		{urn:nbn:de:0030-drops-138231},
  doi =		{10.4230/LIPIcs.SoCG.2021.24},
  annote =	{Keywords: Largest empty rectangle, largest empty box, Klee’s measure problem}
}

Keywords: Largest empty rectangle, largest empty box, Klee’s measure problem
Collection: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue Date: 2021
Date of publication: 02.06.2021

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI