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DOI: 10.4230/LIPIcs.ISAAC.2024.5

Constrained Two-Line Center Problems

Authors: Taehoon Ahn and Sang Won Bae

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

Given a set P of n points in the plane, the two-line center problem asks to find two lines that minimize the maximum distance from each point in P to its closer one of the two resulting lines. The currently best algorithm for the problem takes O(n² log² n) time by Jaromczyk and Kowaluk in 1995. In this paper, we present faster algorithms for three variants of the two-line center problem in which the orientations of the resulting lines are constrained. Specifically, our algorithms solve the problem in O(n log n) time when the orientations of both lines are fixed; in O(n log³ n) time when the orientation of one line is fixed; and in O(n² α(n) log n) time when the angle between the two lines is fixed, where α(n) denotes the inverse Ackermann function.

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Taehoon Ahn and Sang Won Bae. Constrained Two-Line Center Problems. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ahn_et_al:LIPIcs.ISAAC.2024.5,
  author =	{Ahn, Taehoon and Bae, Sang Won},
  title =	{{Constrained Two-Line Center Problems}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{5:1--5:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.5},
  URN =		{urn:nbn:de:0030-drops-221327},
  doi =		{10.4230/LIPIcs.ISAAC.2024.5},
  annote =	{Keywords: two-line center problem, geometric location problem, geometric optimization}
}

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DOI: 10.4230/LIPIcs.SoCG.2022.51

Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

Authors: Mincheol Kim, Chanyang Seo, Taehoon Ahn, and Hee-Kap Ahn

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

We present an algorithm to compute the geodesic L₁ farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes O(nm+n log n + mlog m) construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in O(log(n+m)) time.

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Mincheol Kim, Chanyang Seo, Taehoon Ahn, and Hee-Kap Ahn. Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kim_et_al:LIPIcs.SoCG.2022.51,
  author =	{Kim, Mincheol and Seo, Chanyang and Ahn, Taehoon and Ahn, Hee-Kap},
  title =	{{Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{51:1--51:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.51},
  URN =		{urn:nbn:de:0030-drops-160596},
  doi =		{10.4230/LIPIcs.SoCG.2022.51},
  annote =	{Keywords: Geodesic distance, L₁ metric, farthest-point Voronoi diagram}
}

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Documents authored by Ahn, Taehoon

Constrained Two-Line Center Problems

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Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

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