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DOI: 10.4230/LIPIcs.SoCG.2024.40

Optimal Algorithm for the Planar Two-Center Problem

Authors: Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

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Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue. Optimal Algorithm for the Planar Two-Center Problem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cho_et_al:LIPIcs.SoCG.2024.40,
  author =	{Cho, Kyungjin and Oh, Eunjin and Wang, Haitao and Xue, Jie},
  title =	{{Optimal Algorithm for the Planar Two-Center Problem}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{40:1--40:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.40},
  URN =		{urn:nbn:de:0030-drops-199857},
  doi =		{10.4230/LIPIcs.SoCG.2024.40},
  annote =	{Keywords: two-center, r-coverage, disk coverage, circular hulls}
}

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DOI: 10.4230/LIPIcs.ISAAC.2021.46

Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces

Authors: Kyungjin Cho and Eunjin Oh

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of axis-parallel affine subspaces of dimension Δ that yields an (1+ε)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ, ε and k. This improves the O(n² d)-time algorithm by Eiben et al. [SODA'21] by a factor of n.

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Kyungjin Cho and Eunjin Oh. Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cho_et_al:LIPIcs.ISAAC.2021.46,
  author =	{Cho, Kyungjin and Oh, Eunjin},
  title =	{{Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.46},
  URN =		{urn:nbn:de:0030-drops-154794},
  doi =		{10.4230/LIPIcs.ISAAC.2021.46},
  annote =	{Keywords: k-means clustering, affine subspaces}
}

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Documents authored by Cho, Kyungjin

Optimal Algorithm for the Planar Two-Center Problem

Abstract

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Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces

Abstract

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