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DOI: 10.4230/LIPIcs.SoCG.2026.31

Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems

Authors: Chaeyoon Chung, Anil Maheshwari, and Michiel Smid

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

Given a set S of n points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in S to its closest line. We present a (1+ε)-approximation algorithm for the two-line-center problem that runs in O((n/ε) log (1/ε)) time, which improves the previously best O(nlog n + (n/ε²) log (1/ε) + (1/ε³)log (1/ε))-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first (1+ε)-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in O(n log n) time and show that it is optimal.

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Chaeyoon Chung, Anil Maheshwari, and Michiel Smid. Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chung_et_al:LIPIcs.SoCG.2026.31,
  author =	{Chung, Chaeyoon and Maheshwari, Anil and Smid, Michiel},
  title =	{{Linear-Time (1+\epsilon)-Approximation Algorithms for Two-Line-Center Problems}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.31},
  URN =		{urn:nbn:de:0030-drops-258374},
  doi =		{10.4230/LIPIcs.SoCG.2026.31},
  annote =	{Keywords: Approximation algorithm, two-line-center problem, k-line-center problem, projective clustering, \epsilon-certificate, \epsilon-coreset, width of a point set}
}

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DOI: 10.4230/LIPIcs.ISAAC.2025.21

Covering Weighted Points Using Unit Squares

Authors: Chaeyoon Chung, Jaegun Lee, and Hee-Kap Ahn

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Given a set of n points in d-dimensional space, each assigned a positive weight, we study the problem of finding k axis-parallel unit hypercubes that maximize the total weight of the points contained in their union. In this paper, we present both exact and (1 - ε)-approximation algorithms for the case of k = 2. We present an exact algorithm that runs in O(n²) time in the plane, improving the previous O(n² log² n)-time result. This algorithm generalizes to higher dimensions and larger k in O(n^{dk/2}) time for fixed d and k. We also present a (1 - ε)-approximation algorithm that runs in O(n log min{n, 1/ε} + 1/ε³) time for k = 2 in the plane, improving the best known result. Our approximation algorithm also extends to higher dimensions.

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Chaeyoon Chung, Jaegun Lee, and Hee-Kap Ahn. Covering Weighted Points Using Unit Squares. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chung_et_al:LIPIcs.ISAAC.2025.21,
  author =	{Chung, Chaeyoon and Lee, Jaegun and Ahn, Hee-Kap},
  title =	{{Covering Weighted Points Using Unit Squares}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.21},
  URN =		{urn:nbn:de:0030-drops-249292},
  doi =		{10.4230/LIPIcs.ISAAC.2025.21},
  annote =	{Keywords: Maximum coverage, Unit squares, Approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.8

Tight Bounds on the Number of Closest Pairs in Vertical Slabs

Authors: Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung, Jean-Lou De Carufel, John Iacono, Anil Maheshwari, Saeed Odak, Michiel Smid, and Csaba D. Tóth

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

Let S be a set of n points in ℝ^d, where d ≥ 2 is a constant, and let H₁,H₂,…,H_{m+1} be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly n/m points of S are between any two successive hyperplanes. Let |A(S,m)| be the number of different closest pairs in the {(m+1) choose 2} vertical slabs that are bounded by H_i and H_j, over all 1 ≤ i < j ≤ m+1. We prove tight bounds for the largest possible value of |A(S,m)|, over all point sets of size n, and for all values of 1 ≤ m ≤ n. As a result of these bounds, we obtain, for any constant ε > 0, a data structure of size O(n), such that for any vertical query slab Q, the closest pair in the set Q ∩ S can be reported in O(n^{1/2+ε}) time. Prior to this work, no linear space data structure with sublinear query time was known.

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Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung, Jean-Lou De Carufel, John Iacono, Anil Maheshwari, Saeed Odak, Michiel Smid, and Csaba D. Tóth. Tight Bounds on the Number of Closest Pairs in Vertical Slabs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.WADS.2025.8,
  author =	{Biniaz, Ahmad and Bose, Prosenjit and Chung, Chaeyoon and De Carufel, Jean-Lou and Iacono, John and Maheshwari, Anil and Odak, Saeed and Smid, Michiel and T\'{o}th, Csaba D.},
  title =	{{Tight Bounds on the Number of Closest Pairs in Vertical Slabs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.8},
  URN =		{urn:nbn:de:0030-drops-242391},
  doi =		{10.4230/LIPIcs.WADS.2025.8},
  annote =	{Keywords: closest pair, vertical slab, data structure}
}

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Documents authored by Chung, Chaeyoon

Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems

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Covering Weighted Points Using Unit Squares

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Tight Bounds on the Number of Closest Pairs in Vertical Slabs

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