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Spanner for the 0/1/∞ Weighted Region Problem

Authors Joachim Gudmundsson , Zijin Huang , André van Renssen , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • weighted region problem
  • approximate shortest path
  • spanner

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Abstract

We consider the problem of computing an approximate weighted shortest path in a weighted planar subdivision, with weights assigned from the set {0, 1, ∞}. The subdivision includes zero-cost regions (0-regions) with weight 0 and obstacles with weight ∞, all embedded in a plane with weight 1. In a polygonal domain, where the 0-regions and obstacles are non-overlapping polygons (not necessarily convex) with in total N vertices, we present an algorithm that computes a (1 + ε)-approximate spanner of the input vertices in expected Õ(N/ε³) time, for 0 < ε < 1. Using our spanner, we can compute a (1 + ε)-approximate weighted shortest path between any two points (not necessarily vertices) in Õ(N/ε³) time. Furthermore, we prove that our results more generally apply to non-polygonal convex regions. Using this generalisation, one can approximate the weak partial Fréchet similarity [Buchin et al., 2009] between two polygonal curves in expected Õ(n²/ε²) time, where n is the total number of vertices of the input curves.

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Joachim Gudmundsson, Zijin Huang, André van Renssen, and Sampson Wong. Spanner for the 0/1/∞ Weighted Region Problem. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.33

Author Details

Joachim Gudmundsson
  • The University of Sydney, Australia
Zijin Huang
  • The University of Sydney, Australia
André van Renssen
  • The University of Sydney, Australia
Sampson Wong
  • University of Copenhagen, Denmark

Funding

  • Gudmundsson, Joachim: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • van Renssen, André: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).

References

  1. Lyudmil Aleksandrov, Anil Maheshwari, and Jörg-Rüdiger Sack. Approximation algorithms for geometric shortest path problems. In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, pages 286-295, New York, NY, USA, May 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335339.
  2. Lyudmil Aleksandrov, Anil Maheshwari, and Jörg-Rüdiger Sack. Determining approximate shortest paths on weighted polyhedral surfaces. Journal of the ACM, 52(1):25-53, January 2005. URL: https://doi.org/10.1145/1044731.1044733.
  3. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 05:75-91, March 1995. URL: https://doi.org/10.1142/S0218195995000064.
  4. Prosenjit Bose, Jean-Lou de Carufel, Pat Morin, André van Renssen, and Sander Verdonschot. Towards tight bounds on theta-graphs: more is not always better. Theoretical Computer Science, 616:70-93, February 2016. URL: https://doi.org/10.1016/j.tcs.2015.12.017.
  5. Prosenjit Bose, Guillermo Esteban, David Orden, and Rodrigo I. Silveira. On approximating shortest paths in weighted triangular tessellations. Artificial Intelligence, 318:103898, May 2023. URL: https://doi.org/10.1016/j.artint.2023.103898.
  6. Prosenjit Bose and André van Renssen. Spanning properties of Yao and theta-graphs in the presence of constraints. International Journal of Computational Geometry & Applications, 29(02):95-120, June 2019. URL: https://doi.org/10.1142/S021819591950002X.
  7. Kevin Buchin, Maike Buchin, and Yusu Wang. Exact algorithms for partial curve matching via the Fréchet distance. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 645-654, USA, January 2009. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611973068.71.
  8. Kevin Buchin, Tim Ophelders, and Bettina Speckmann. SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2887-2901, USA, January 2019. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611975482.179.
  9. Siu-Wing Cheng, Jiongxin Jin, and Antoine Vigneron. Triangulation Refinement and Approximate Shortest Paths in Weighted Regions. In Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pages 1626-1640. Society for Industrial and Applied Mathematics, December 2014. URL: https://doi.org/10.1137/1.9781611973730.108.
  10. Kenneth L. Clarkson. Approximation algorithms for shortest path motion planning. In Proceedings of the nineteenth annual ACM Symposium on Theory of Computing, pages 56-65, New York, NY, USA, January 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28402.
  11. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer, Berlin, Heidelberg, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  12. Sarita de Berg, Guillermo Esteban, Rodrigo I. Silveira, and Frank Staals. Exact solutions to the Weighted Region Problem, February 2024. arXiv:2402.12028 [cs]. URL: https://doi.org/10.48550/arXiv.2402.12028.
  13. Jean-Lou de Carufel, Amin Gheibi, Anil Maheshwari, Jörg-Rüdiger Sack, and Christian Scheffer. Similarity of polygonal curves in the presence of outliers. Computational Geometry, 47(5):625-641, July 2014. URL: https://doi.org/10.1016/j.comgeo.2014.01.002.
  14. Jean-Lou de Carufel, Carsten Grimm, Anil Maheshwari, Megan Owen, and Michiel Smid. A note on the unsolvability of the weighted region shortest path problem. Computational Geometry, 47(7):724-727, August 2014. URL: https://doi.org/10.1016/j.comgeo.2014.02.004.
  15. Laxmi P. Gewali, Alex C. Meng, Joseph S. B. Mitchell, and Simeon Ntafos. Path planning in 0/1/∞ weighted regions with applications. In Proceedings of the Fourth Annual Symposium on Computational Geometry, pages 266-278, New York, NY, USA, January 1988. Association for Computing Machinery. URL: https://doi.org/10.1145/73393.73421.
  16. Mark Lanthier, Anil Maheshwari, and Jörg-Rüdiger Sack. Approximating weighted shortest paths on polyhedral surfaces. In Proceedings of the Thirteenth Annual Symposium on Computational Geometry, pages 274-283, New York, NY, USA, August 1997. Association for Computing Machinery. URL: https://doi.org/10.1145/262839.262984.
  17. Joseph S. B. Mitchell and Christos H. Papadimitriou. The weighted region problem: finding shortest paths through a weighted planar subdivision. Journal of the ACM, 38(1):18-73, January 1991. URL: https://doi.org/10.1145/102782.102784.
  18. Zheng Sun and John H. Reif. On finding approximate optimal paths in weighted regions. Journal of Algorithms, 58(1):1-32, January 2006. URL: https://doi.org/10.1016/j.jalgor.2004.07.004.
  19. Victor Andreevich Toponogov. Differential Geometry of Curves and Surfaces: A Concise Guide. Birkhauser, 2006. Google Scholar
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