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.2023.24
URN: urn:nbn:de:0030-drops-178741
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17874/
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Chan, Timothy M.

Minimum L_∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem

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LIPIcs-SoCG-2023-24.pdf (0.7 MB)

Abstract

We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem.
To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis.

BibTeX - Entry

@InProceedings{chan:LIPIcs.SoCG.2023.24,
  author =	{Chan, Timothy M.},
  title =	{{Minimum L\underline∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17874},
  URN =		{urn:nbn:de:0030-drops-178741},
  doi =		{10.4230/LIPIcs.SoCG.2023.24},
  annote =	{Keywords: Hausdorff distance, geometric optimization, Klee’s measure problem, fine-grained complexity}
}

Keywords: Hausdorff distance, geometric optimization, Klee’s measure problem, fine-grained complexity
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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