Document Open Access Logo

An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

Authors Bruce W. Brewer , Haitao Wang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.31.pdf
  • Filesize: 0.69 MB
  • 8 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • disk graphs
  • weighted Voronoi diagrams
  • shortest paths

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Given in the plane a set S of n points and a set of disks centered at these points, the disk graph G(S) induced by these disks has vertex set S and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point s ∈ S to all vertices in G(S) where the length of a path in G(S) is defined as the number of edges in the path. The previously best algorithm solves the problem in O(nlog² n) time. A lower bound of Ω(nlog n) is also known for this problem under the algebraic decision tree model. In this paper, we present an O(nlog n) time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

Cite As Get BibTex

Bruce W. Brewer and Haitao Wang. An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 31:1-31:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.31

Author Details

Bruce W. Brewer
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA

References

  1. Shinwoo An, Kyungjin Cho, and Eunjin Oh. Faster algorithms for cycle hitting problems on disk graphs. In Proceedings of the 18th Algorithms and Data Structures Symposium (WADS), pages 29-42, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_3.
  2. Shinwoo An, Eunjin Oh, and Jie Xue. Single-source shortest path problem in weighted disk graphs. In Proceedings of the 41st International Symposium on Computational Geometry (SoCG), pages 7:1-7:15, 2025. URL: https://doi.org/10.4230/LIPIcs.SoCG.2025.7.
  3. Alexander Baumann, Haim Kaplan, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic connectivity in disk graphs. Discrete and Computational Geometry, 71:214-277, 2024. URL: https://doi.org/10.1007/s00454-023-00621-x.
  4. Mark de Berg and Sergio Cabello. An O(n log n) algorithm for single-source shortest paths in disk graphs. In Proceedings of the 33rd Annual European Symposium on Algorithms (ESA), page to appear, 2025. URL: https://doi.org/10.48550/arXiv.2506.07571.
  5. Bruce W. Brewer and Haitao Wang. An improved algorithm for shortest paths in weighted unit-disk graphs. In Proceedings of the 36th Canadian Conference on Computational Geometry (CCCG), pages 57-64, 2024. URL: https://arxiv.org/abs/2407.03176.
  6. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry: Theory and Applications, 48:360-367, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.12.003.
  7. Sergio Cabello and Wolfgang Mulzer. Minimum cuts in geometric intersection graphs. Computational Geometry: Theory and Applications, 94(101720), 2021. URL: https://doi.org/10.1016/j.comgeo.2020.101720.
  8. Timothy M. Chan and Zhengcheng Huang. Faster algorithms for reverse shortest path in unit-disk graphs and related geometric optimization problems: Improving the shrink-and-bifurcate technique. In Proceedings of the 41st International Symposium on Computational Geometry (SoCG), pages 32:1-32:14, 2025. URL: https://doi.org/10.4230/LIPIcs.SoCG.2025.32.
  9. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.24.
  10. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86:165-177, 1990. URL: https://doi.org/10.1016/0012-365X(90)90358-O.
  11. Herbert Edelsbrunner, Leonidas J. Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986. URL: https://doi.org/10.1137/0215023.
  12. Herbert Edelsbrunner and Ernst P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9:66-104, 1990. URL: https://doi.org/10.1145/77635.77639.
  13. Jared Espenant, J. Mark Keil, and Debajyoti Mondal. Finding a maximum clique in a disk graph. In Proceedings of the 39th International Symposium on Computational Geometry (SoCG), pages 30:1-30:17, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.49.
  14. Steven Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987. URL: https://doi.org/10.1007/BF01840357.
  15. Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir. The unweighted and weighted reverse shortest path problem for disk graphs. In Proceedings of the 31st Annual European Symposium on Algorithms (ESA), pages 67:1-67:14, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.67.
  16. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. Discrete and Computational Geometry, 64:838-904, 2020. URL: https://doi.org/10.1007/s00454-020-00243-7.
  17. David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12:28-35, 1983. URL: https://doi.org/10.1137/0212002.
  18. Katharina Klost. An algorithmic framework for the single source shortest path problem with applications to disk graphs. Computational Geometry: Theory and Applications, 111:101979, 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101979.
  19. Chih-Hung Liu. Nearly optimal planar k nearest neighbors queries under general distance functions. SIAM Journal on Computing, 51:723-765, 2022. URL: https://doi.org/10.1137/20M1388371.
  20. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Subexponential parameterized algorithms on disk graphs (extended abstract). In Proceedings of the 33rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2005-2031, 2022. URL: https://doi.org/10.1137/1.9781611977073.80.
  21. Evanthia Papadopoulou and D.T. Lee. The L_∞-Voronoi diagram of segments and VLSI applications. International Journal of Computational Geometry and Application, 11(101720):503-528, 2001. URL: https://doi.org/10.1142/S0218195901000626.
  22. Liam Roditty and Michael Segal. On bounded leg shortest paths problems. Algorithmica, 59:583-600, 2011. URL: https://doi.org/10.1007/s00453-009-9322-3.
  23. Micha Sharir. Intersection and closest-pair problems for a set of planar discs. SIAM Journal on Computing, 14:448-468, 1985. URL: https://doi.org/10.1137/0214034.
  24. Anastasiia Tkachenko and Haitao Wang. Dominating set, independent set, discrete k-center, dispersion, and related problems for planar points in convex position. In Proceedings of the 42nd International Symposium on Theoretical Aspects of Computer Science (STACS), pages 73:1-73:20, 2025. URL: https://doi.org/10.4230/LIPIcs.STACS.2025.73.
  25. Haitao Wang and Jie Xue. Near-optimal algorithms for shortest paths in weighted unit-disk graphs. Discrete and Computational Geometry, 64:1141-1166, 2020. URL: https://doi.org/10.1007/s00454-020-00219-7.
  26. C.-K. Yap. A geometric consistency theorem for a symbolic perturbation scheme. Journal of Computer and System Sciences, 40:2-18, 1990. URL: https://doi.org/10.1016/0022-0000(90)90016-E.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact