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The Rectilinear Convex Hull of Line Segments

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Fundamentals of Computation Theory (FCT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14292))

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Abstract

We explore an extension to rectilinear convexity of the classic problem of computing the convex hull of a collection of line segments. Namely, we solve the problem of computing and maintaining the rectilinear convex hull of a set of n line segments, while we simultaneously rotate the coordinate axes by an angle that goes from 0 to \(2\pi \).

We describe an algorithm that runs in optimal \(\varTheta (n\log n)\) time and \(\varTheta (n\alpha (n))\) space for segments that are non-necessarily disjoint, where \(\alpha (n)\) is the inverse of the Ackermann’s function. If instead the line segments form a simple polygonal chain, the algorithm can be adapted so as to improve the time and space complexities to \(\varTheta (n)\).

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Notes

  1. 1.

    In the literature, orthogonal convexity is also known as ortho-convexity [19] or x-y convexity [14].

  2. 2.

    See the definition of the maximal r-convex hull in [15].

  3. 3.

    In the literature, \(\mathcal {O}\)-convexity is also known as D-convexity [20], Directional convexity [8], Set-theoretical D-convexity [9], or Partial convexity [13].

  4. 4.

    We remark that, since the set \(\mathcal {O}\) is formed by two orthogonal lines parallel to the coordinate axis, in this paper \(\mathcal {O}\)-convexity is equivalent to Orthogonal Convexity.

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Acknowledgements

Justin Dallant is supported by the French Community of Belgium via the funding of a FRIA grant. Pablo Pérez-Lantero was partially supported by project DICYT 042332PL Vicerrectoría de Investigación, Desarrollo e Innovación USACH (Chile). Carlos Seara is supported by Project PID2019-104129GB-I00/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università Roma Tre, Rome, Italy

    Carlos Alegría

  2. Computer Science Department, Faculté des Sciences, Université libre de Bruxelles, Bruxelles, Belgium

    Justin Dallant

  3. Departamento de Matemática y Computación, Universidad de Santiago de Chile, Santiago, Chile

    Pablo Pérez-Lantero

  4. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Carlos Seara

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  1. Carlos Alegría
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  2. Justin Dallant
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  4. Carlos Seara
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Correspondence to Carlos Seara .

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Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

  2. University of Kiel, Kiel, Germany

    Klaus Jansen

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Alegría, C., Dallant, J., Pérez-Lantero, P., Seara, C. (2023). The Rectilinear Convex Hull of Line Segments. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham. https://doi.org/10.1007/978-3-031-43587-4_3

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  • DOI: https://doi.org/10.1007/978-3-031-43587-4_3

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