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Minimum-Width Double-Slabs and Widest Empty Slabs in High Dimensions

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LATIN 2024: Theoretical Informatics (LATIN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14578))

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Abstract

A slab in d-dimensional space \(\mathbb {R}^d\) is the set of points enclosed by two parallel hyperplanes. We consider the problem of finding an optimal pair of parallel slabs, called a double-slab, that covers a given set P of n points in \(\mathbb {R}^d\). We address two optimization problems in \(\mathbb {R}^d\) for any fixed dimension \(d\geqslant 3\): the minimum-width double-slab problem, in which one wants to minimize the maximum width of the two slabs of the resulting double-slab, and the widest empty slab problem, in which one wants to maximize the gap between the two slabs. Our results include the first nontrivial exact algorithms that solve the former problem for \(d\geqslant 3\) and the latter problem for \(d\geqslant 4\).

C. Chung, T. Ahn, and H.-K. Ahn were supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)) and (No. 2019-0-01906, Artificial Intelligence Graduate School Program(POSTECH)). T. Ahn, S.W. Bae, and C. Chung were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2023-00251168).

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Notes

  1. 1.

    Note that the relative interior of a 0-flat (a point) is by definition the point itself.

  2. 2.

    In fact they construct a vertical decomposition of the entire arrangement.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, Korea

    Taehoon Ahn & Chaeyoon Chung

  2. Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, Korea

    Hee-Kap Ahn

  3. Division of Computer Science and Engineering, Kyonggi University, Suwon, Korea

    Sang Won Bae

  4. SCALGO, Aarhus, Denmark

    Otfried Cheong

  5. Department of Service and Design Engineering, Sungshin Women’s University, Seoul, Korea

    Sang Duk Yoon

Authors
  1. Taehoon Ahn
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  2. Chaeyoon Chung
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  3. Hee-Kap Ahn
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  4. Sang Won Bae
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  5. Otfried Cheong
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  6. Sang Duk Yoon
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Corresponding author

Correspondence to Sang Won Bae .

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Editors and Affiliations

  1. DIM-CMM, Universidad de Chile, Santiago, Chile

    José A. Soto

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

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Ahn, T., Chung, C., Ahn, HK., Bae, S.W., Cheong, O., Yoon, S.D. (2024). Minimum-Width Double-Slabs and Widest Empty Slabs in High Dimensions. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14578. Springer, Cham. https://doi.org/10.1007/978-3-031-55598-5_20

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  • DOI: https://doi.org/10.1007/978-3-031-55598-5_20

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