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On the Complexity of Closest Pair via Polar-Pair of Point-Sets

Authors: Roee David, Karthik C. S., and Bundit Laekhanukit

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)


Abstract
Every graph G can be represented by a collection of equi-radii spheres in a d-dimensional metric Delta such that there is an edge uv in G if and only if the spheres corresponding to u and v intersect. The smallest integer d such that G can be represented by a collection of spheres (all of the same radius) in Delta is called the sphericity of G, and if the collection of spheres are non-overlapping, then the value d is called the contact-dimension of G. In this paper, we study the sphericity and contact dimension of the complete bipartite graph K_{n,n} in various L^p-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.

Cite as

Roee David, Karthik C. S., and Bundit Laekhanukit. On the Complexity of Closest Pair via Polar-Pair of Point-Sets. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{david_et_al:LIPIcs.SoCG.2018.28,
  author =	{David, Roee and C. S., Karthik and Laekhanukit, Bundit},
  title =	{{On the Complexity of Closest Pair via Polar-Pair of Point-Sets}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{28:1--28:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.28},
  URN =		{urn:nbn:de:0030-drops-87412},
  doi =		{10.4230/LIPIcs.SoCG.2018.28},
  annote =	{Keywords: Contact dimension, Sphericity, Closest Pair, Fine-Grained Complexity}
}
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