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Faster Algorithms for Largest Empty Rectangles and Boxes

Author Timothy M. Chan

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Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

Supported in part by NSF Grant CCF-1814026.

Acknowledgements

I thank David Zheng for discussions on the 2D problem.

Cite As Get BibTex

Timothy M. Chan. Faster Algorithms for Largest Empty Rectangles and Boxes. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.24

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Largest empty rectangle
  • largest empty box
  • Klee’s measure problem

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