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An Improved Bound for Plane Covering Paths

Authors Hugo A. Akitaya , Greg Aloupis, Ahmad Biniaz , Prosenjit Bose , Jean-Lou De Carufel , Cyril Gavoille , John Iacono , Linda Kleist , Michiel Smid , Diane Souvaine , Leonidas Theocharous

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Covering Path
  • Upper Bound
  • Simple Algorithm

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Abstract

A covering path for a finite set P of points in the plane is a polygonal path such that every point of P lies on a segment of the path. The vertices of the path need not be at points of P. A covering path is plane if its segments do not cross each other. Let π(n) be the minimum number such that every set of n points in the plane admits a plane covering path with at most π(n) segments. We prove that π(n) ≤  ⌈6n/7⌉. This improves the previous best-known upper bound of ⌈21n/22⌉, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple O(n log n)-time algorithm for computing a plane covering path.

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Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, and Leonidas Theocharous. An Improved Bound for Plane Covering Paths. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 75:1-75:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.75

Author Details

Hugo A. Akitaya
  • Miner School of Computer & Information Sciences, University of Massachusetts, Lowell, MA, USA
Greg Aloupis
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA
Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Cyril Gavoille
  • LaBRI, University of Bordeaux, France
John Iacono
  • Department of Computer Science, Université libre de Bruxelles, Belgium
Linda Kleist
  • Department of Computer Science, Universität Potsdam, Germany
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada
Diane Souvaine
  • Department of Computer Science, Tufts University, Medford, MA, USA
Leonidas Theocharous
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada

Funding

  • A. Akitaya, Hugo: Supported by the NSF award CCF-2348067.
  • Biniaz, Ahmad: Supported in part by NSERC.
  • Bose, Prosenjit: Supported in part by NSERC.
  • De Carufel, Jean-Lou: Supported in part by NSERC.
  • Gavoille, Cyril: Supported by the French ANR projects ANR-22-CE48-0001 (TEMPOGRAL) and ANR-24-CE48-7768-01 (ENEDISC).
  • Iacono, John: This work was supported by the Fonds de la Recherche Scientifique-FNRS.
  • Smid, Michiel: Supported in part by NSERC.
  • Theocharous, Leonidas: Supported in part by NSERC.

Acknowledgements

This work was initiated at the 12th Annual Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Holetown, Barbados, in February 2025. The authors thank the organizers and participants.

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