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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

We consider two optimization problems of approximating a convex polygon, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation θ is a copy of an axis-aligned histogon rotated by θ in counterclockwise direction. The goal is to find a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0,π). Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.

Jaehoon Chung, Sang Won Bae, Chan-Su Shin, Sang Duk Yoon, and Hee-Kap Ahn. Inscribing or Circumscribing a Histogon to a Convex Polygon. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chung_et_al:LIPIcs.FSTTCS.2022.13, author = {Chung, Jaehoon and Bae, Sang Won and Shin, Chan-Su and Yoon, Sang Duk and Ahn, Hee-Kap}, title = {{Inscribing or Circumscribing a Histogon to a Convex Polygon}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.13}, URN = {urn:nbn:de:0030-drops-174054}, doi = {10.4230/LIPIcs.FSTTCS.2022.13}, annote = {Keywords: Shape simplification, Shape analysis, Histogon, Convex polygon} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

This paper studies empty squares in arbitrary orientation among a set P of n points in the plane. We prove that the number of empty squares with four contact pairs is between Ω(n) and O(n²), and that these bounds are tight, provided P is in a certain general position. A contact pair of a square is a pair of a point p ∈ P and a side 𝓁 of the square with p ∈ 𝓁. The upper bound O(n²) also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. We then present an algorithm that maintains a combinatorial structure of the L_∞ Voronoi diagram of P, while the axes of the plane continuously rotate by 90 degrees, and simultaneously reports all empty squares with four contact pairs among P in an output-sensitive way within O(slog n) time and O(n) space, where s denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among P and a square annulus of minimum width or minimum area that encloses P over all orientations can be computed in worst-case O(n² log n) time.

Sang Won Bae and Sang Duk Yoon. Empty Squares in Arbitrary Orientation Among Points. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bae_et_al:LIPIcs.SoCG.2020.13, author = {Bae, Sang Won and Yoon, Sang Duk}, title = {{Empty Squares in Arbitrary Orientation Among Points}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.13}, URN = {urn:nbn:de:0030-drops-121716}, doi = {10.4230/LIPIcs.SoCG.2020.13}, annote = {Keywords: empty square, arbitrary orientation, Erd\H{o}s - Szekeres problem, L\underline∞ Voronoi diagram, largest empty square problem, square annulus} }

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

We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, and Sang Duk Yoon. The Reverse Kakeya Problem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bae_et_al:LIPIcs.SoCG.2018.6, author = {Bae, Sang Won and Cabello, Sergio and Cheong, Otfried and Choi, Yoonsung and Stehn, Fabian and Yoon, Sang Duk}, title = {{The Reverse Kakeya Problem}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {6:1--6:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.6}, URN = {urn:nbn:de:0030-drops-87199}, doi = {10.4230/LIPIcs.SoCG.2018.6}, annote = {Keywords: Kakeya problem, convex, isodynamic point, turning} }

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