Document Open Access Logo

Link Diameter, Radius and 2-Point Link Distance Queries in Polygonal Domains

Authors Mart Hagedoorn , Valentin Polishchuk

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.WADS.2025.34.pdf
  • Filesize: 1.07 MB
  • 15 pages
HTML5 Logo HTML (experimental)
LIPIcs.WADS.2025.34.html

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Minimum-link paths
  • link distance
  • diameter
  • center
  • radius
  • 2-point distance queries

Metrics

Abstract

We show how to preprocess a polygonal domain with holes so that the link distance (the number of links in a minimum-link path) between two query points in the domain can be reported efficiently. Using our data structures, the link diameter of the domain (i.e., the maximum number of links that may be required in a minimum-link path between two points in the domain) as well as the link center and radius of the domain (i.e., the point minimizing the maximum link distance to the furthest point in the domain and this maximum link distance) can be found in polynomial time. We also give a simpler algorithm for finding the link diameter, not using the link distance query structures. Answering 2-point link distance queries and computing the link diameter/radius/center in polygonal domains have been open questions since these problems were studied for simple polygons in the 90’s.

Cite As Get BibTex

Mart Hagedoorn and Valentin Polishchuk. Link Diameter, Radius and 2-Point Link Distance Queries in Polygonal Domains. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.34

Author Details

Mart Hagedoorn
  • Technische Universität Dortmund, Germany
Valentin Polishchuk
  • Linköping University, Sweden

References

  1. Mikkel Abrahamsen, Joakim Blikstad, André Nusser, and Hanwen Zhang. Minimum star partitions of simple polygons in polynomial time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 904-910, 2024. URL: https://doi.org/10.1145/3618260.3649756.
  2. Alok Aggarwal, Heather Booth, Joseph O'Rourke, Subhash Suri, and Chee K Yap. Finding minimal convex nested polygons. Information and Computation, 83(1):98-110, 1989. URL: https://doi.org/10.1016/0890-5401(89)90049-7.
  3. Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A linear-time algorithm for the geodesic center of a simple polygon. Discrete & Computational Geometry, 56(4):836-859, 2016. URL: https://doi.org/10.1007/s00454-016-9796-0.
  4. Esther M Arkin, Joseph SB Mitchell, and Subhash Suri. Logarithmic-time link path queries in a simple polygon. International Journal of Computational Geometry & Applications, 5(04):369-395, 1995. URL: https://doi.org/10.1142/S0218195995000234.
  5. Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André Van Renssen, and Marcel Roeloffzen. Rectilinear link diameter and radius in a rectilinear polygonal domain. Computational Geometry, 92:101685, 2021. URL: https://doi.org/10.1016/j.comgeo.2020.101685.
  6. Sang Won Bae, Matias Korman, and Yoshio Okamoto. The geodesic diameter of polygonal domains. Discrete & Computational Geometry, 50(2):306-329, 2013. URL: https://doi.org/10.1007/s00454-013-9527-8.
  7. Sang Won Bae, Matias Korman, and Yoshio Okamoto. Computing the geodesic centers of a polygonal domain. Computational Geometry, 77:3-9, 2019. URL: https://doi.org/10.1016/j.comgeo.2015.10.009.
  8. Danny Z Chen and Haitao Wang. Visibility and ray shooting queries in polygonal domains. Computational Geometry, 48(2):31-41, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.08.003.
  9. Yi-Jen Chiang and Joseph SB Mitchell. Two-point Euclidean shortest path queries in the plane. In Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, pages 215-224. Citeseer, 1999. URL: https://doi.org/10.5555/314500.314560.
  10. Sarita de Berg, Tillmann Miltzow, and Frank Staals. Towards space efficient two-point shortest path queries in a polygonal domain. In 40th International Symposium on Computational Geometry (SoCG 2024), volume 293, pages 17:1-17:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.17.
  11. Hristo N Djidjev, Andrzej Lingas, and Jörg-Rüdiger Sack. An O(nlog n) algorithm for computing the link center of a simple polygon. Discrete & Computational Geometry, 8(2):131-152, 1992. URL: https://doi.org/10.1007/BF02293040.
  12. 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.
  13. Stephan J Eidenbenz and Peter Widmayer. An approximation algorithm for minimum convex cover with logarithmic performance guarantee. SIAM Journal on Computing, 32(3):654-670, 2003. URL: https://doi.org/10.1137/S0097539702405139.
  14. Anka Gajentaan and Mark H Overmars. On a class of O(n²) problems in computational geometry. Computational geometry, 5(3):165-185, 1995. URL: https://doi.org/10.1016/0925-7721(95)00022-2.
  15. Subir Kumar Ghosh. Computing the visibility polygon from a convex set and related problems. Journal of Algorithms, 12(1):75-95, 1991. URL: https://doi.org/10.1016/0196-6774(91)90024-S.
  16. Leonidas J Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. In Proceedings of the third annual symposium on Computational geometry, pages 50-63, 1987. URL: https://doi.org/10.1145/41958.41964.
  17. John Hershberger and Jack Snoeyink. Computing minimum length paths of a given homotopy class. Computational geometry, 4(2):63-97, 1994. URL: https://doi.org/10.1016/0925-7721(94)90010-8.
  18. John Hershberger and Subhash Suri. Matrix searching with the shortest-path metric. SIAM Journal on Computing, 26(6):1612-1634, 1997. URL: https://doi.org/10.1137/S0097539793253577.
  19. Simon Kahan and Jack Snoeyink. On the bit complexity of minimum link paths: Superquadratic algorithms for problems solvable in linear time. Computational Geometry, 12(1):33-44, 1999. URL: https://doi.org/10.1016/S0925-7721(98)00041-8.
  20. Yan Ke. An efficient algorithm for link-distance problems. In Proceedings of the fifth annual symposium on Computational geometry, pages 69-78, 1989. URL: https://doi.org/10.1145/73833.73841.
  21. Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk, and Frank Staals. On the complexity of minimum-link path problems. Journal of Computational Geometry, 8(2):80-108, 2017. Special issue on SoCG'16. URL: https://doi.org/10.20382/jocg.v8i2a5.
  22. Joseph SB Mitchell, C Piatko, and Esther M Arkin. Computing a shortest k-link path in a polygon. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 573-582. IEEE, 1992. URL: https://doi.org/10.1109/sfcs.1992.267794.
  23. Jörg-Rüdiger Sack and Jorge Urrutia. Handbook of computational geometry. Elsevier, 1999. Google Scholar
  24. Joseph SB Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-link paths revisited. Computational Geometry, 47(6):651-667, 2014. Special issue on EuroCG'11. URL: https://doi.org/10.1016/J.COMGEO.2013.12.005.
  25. Joseph SB Mitchell, Günter Rote, and Gerhard J Woeginger. Minimum-link paths among obstacles in the plane. Algorithmica, 8(1):431-459, 1992. URL: https://doi.org/10.1007/bf01758855.
  26. Subhash Suri. A linear time algorithm for minimum link paths inside a simple polygon. Computer Vision, Graphics, and Image Processing, 35(1):99-110, 1986. URL: https://doi.org/10.1016/0734-189x(86)90070-8.
  27. Subhash Suri. Computing geodesic furthest neighbors in simple polygons. Journal of Computer and System Sciences, 39(2):220-235, 1989. URL: https://doi.org/10.1016/0022-0000(89)90045-7.
  28. Subhash Suri. On some link distance problems in a simple polygon. IEEE transactions on Robotics and Automation, 6(1):108-113, 1990. URL: https://doi.org/10.1109/70.88124.
  29. Subhash Suri and Joseph O'Rourke. Worst-case optimal algorithms for constructing visibility polygons with holes. In Proceedings of the second annual symposium on Computational geometry, pages 14-23, 1986. URL: https://doi.org/10.1145/10515.10517.
  30. Csaba D Toth, Joseph O'Rourke, and Jacob E Goodman. Handbook of discrete and computational geometry. CRC press, 2017. URL: https://doi.org/10.1201/9781315119601.
  31. Haitao Wang. On the geodesic centers of polygonal domains. J. Comput. Geom., 9(1):131-190, 2018. URL: https://doi.org/10.20382/jocg.v9i1a5.
  32. Haitao Wang. A divide-and-conquer algorithm for two-point L₁ shortest path queries in polygonal domains. Journal of Computational Geometry, 11(1):235-282, 2020. URL: https://doi.org/10.20382/jocg.v11i1a10.
  33. Haitao Wang. A new algorithm for Euclidean shortest paths in the plane. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 975-988, 2021. URL: https://doi.org/10.1145/3406325.3451037.
  34. Haitao Wang. Shortest paths among obstacles in the plane revisited. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, pages 810-821, 2021. URL: https://doi.org/10.1137/1.9781611976465.51.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

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