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DOI: 10.4230/LIPIcs.ESA.2021.54

Certified Approximation Algorithms for the Fermat Point and n-Ellipses

Authors: Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract

Given a set A of n points in ℝ^d with weight function w: A→ℝ_{> 0}, the Fermat distance function is φ(x): = ∑_{a∈A}w(a)‖x-a‖. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function φ. We consider the problem of computing a point x̃* that is an ε-approximation of x* in the sense that ‖x̃*-x*‖<ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x*). We devise a certified subdivision algorithm for computing x̃*, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x*, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ^{-1}(r) for r > φ(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques.

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Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Certified Approximation Algorithms for the Fermat Point and n-Ellipses. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{junginger_et_al:LIPIcs.ESA.2021.54,
  author =	{Junginger, Kolja and Mantas, Ioannis and Papadopoulou, Evanthia and Suderland, Martin and Yap, Chee},
  title =	{{Certified Approximation Algorithms for the Fermat Point and n-Ellipses}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{54:1--54:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.54},
  URN =		{urn:nbn:de:0030-drops-146359},
  doi =		{10.4230/LIPIcs.ESA.2021.54},
  annote =	{Keywords: Fermat point, n-ellipse, subdivision, approximation, certified, algorithms}
}

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DOI: 10.4230/LIPIcs.SoCG.2018.50

Deletion in Abstract Voronoi Diagrams in Expected Linear Time

Authors: Kolja Junginger and Evanthia Papadopoulou

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

Abstract

Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed version of a Voronoi construct that has a structure similar to an abstract Voronoi diagram, without however being one. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under an insertion operation, thus, enabling its use in incremental constructions.

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Kolja Junginger and Evanthia Papadopoulou. Deletion in Abstract Voronoi Diagrams in Expected Linear Time. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{junginger_et_al:LIPIcs.SoCG.2018.50,
  author =	{Junginger, Kolja and Papadopoulou, Evanthia},
  title =	{{Deletion in Abstract Voronoi Diagrams in Expected Linear Time}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{50:1--50:14},
  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.2018.50},
  URN =		{urn:nbn:de:0030-drops-87639},
  doi =		{10.4230/LIPIcs.SoCG.2018.50},
  annote =	{Keywords: Abstract Voronoi diagram, linear-time algorithm, update after deletion, randomized incremental algorithm}
}

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Documents authored by Junginger, Kolja

Certified Approximation Algorithms for the Fermat Point and n-Ellipses

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Deletion in Abstract Voronoi Diagrams in Expected Linear Time

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