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DOI: 10.4230/LIPIcs.ESA.2026.44

Smallest Convex Hulls of Polygons

Authors: Mook Kwon Jung and Hee-Kap Ahn

Published in: LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026)

Abstract

We study the problem of minimizing the area of the convex hull of k polygons with a total of n vertices in the plane, under translations and rigid motions for any fixed k ≥ 3. For any ε ∈ (0, 1), we give (1 + ε)-approximation algorithms running in O(ε^{-1/2} log n + ε^{1/2 - k}) time for translations, and in O(ε^{-1/2} log n + ε^{3/2 - 2k}) time for rigid motions. We also consider minimizing the perimeter of the convex hull under translations and obtain a (1 + ε)-approximation algorithm running in O(ε^{-1/2} log n + ε^{1/2 - k}log^{k-1}(1/ε)) time. To the best of our knowledge, these are the first results of this kind for k ≥ 3 polygons. Furthermore, for the special case of two polygons with n₀ and n₁ vertices (n₀ ≥ n₁), respectively, we give an O(n₀+n₁log²(n₀+n₁))-time algorithm for the minimum-perimeter problem. This significantly improves upon the best-known O((n₀+n₁)log²(n₀+n₁)) bound by eliminating the logarithmic overhead associated with the larger input size n₀.

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Mook Kwon Jung and Hee-Kap Ahn. Smallest Convex Hulls of Polygons. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jung_et_al:LIPIcs.ESA.2026.44,
  author =	{Jung, Mook Kwon and Ahn, Hee-Kap},
  title =	{{Smallest Convex Hulls of Polygons}},
  booktitle =	{34th Annual European Symposium on Algorithms (ESA 2026)},
  pages =	{44:1--44:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-445-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{388},
  editor =	{Bille, Philip and Pettie, Seth and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.44},
  URN =		{urn:nbn:de:0030-drops-271803},
  doi =		{10.4230/LIPIcs.ESA.2026.44},
  annote =	{Keywords: Convex hull, packing, bundling, approximation algorithms, cuttings}
}

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DOI: 10.4230/LIPIcs.WADS.2025.48

Farthest-Point Voronoi Diagrams in the Hilbert Metric

Authors: Minju Song, Mook Kwon Jung, and Hee-Kap Ahn

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

Abstract

The Hilbert metric, introduced by David Hilbert in 1895, is a projective metric defined on a bounded convex domain in a Euclidean space. For a convex polygon with m vertices and n point sites lying inside the polygon in the plane, it is shown that the nearest-point Voronoi diagram in the Hilbert metric has combinatorial complexity of O(mn) [Gezalyan and Mount, SoCG 2023]. In this paper, we show that the farthest-point Voronoi diagram in the Hilbert metric has combinatorial complexity O(m), which is independent of the number of sites. Also, we present an efficient algorithm to compute the farthest-point Voronoi diagram.

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Minju Song, Mook Kwon Jung, and Hee-Kap Ahn. Farthest-Point Voronoi Diagrams in the Hilbert Metric. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{song_et_al:LIPIcs.WADS.2025.48,
  author =	{Song, Minju and Jung, Mook Kwon and Ahn, Hee-Kap},
  title =	{{Farthest-Point Voronoi Diagrams in the Hilbert Metric}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{48:1--48:15},
  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.2025.48},
  URN =		{urn:nbn:de:0030-drops-242797},
  doi =		{10.4230/LIPIcs.WADS.2025.48},
  annote =	{Keywords: Farthest-point Voronoi diagram, Hilbert metric, Complexity, Algorithm}
}

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Documents authored by Jung, Mook Kwon

Smallest Convex Hulls of Polygons

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Farthest-Point Voronoi Diagrams in the Hilbert Metric

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