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A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon

Authors Mark de Berg , Leonidas Theocharous

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Author Details

Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • de Berg, Mark: MdB is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.
  • Theocharous, Leonidas: LT is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

Cite As Get BibTex

Mark de Berg and Leonidas Theocharous. A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.16

Abstract

Let 𝒫 be a simple polygon with m vertices and let P be a set of n points inside 𝒫. We prove that there exists, for any ε > 0, a set C ⊂ P of size O(1/ε²) such that the following holds: for any query point q inside the polygon 𝒫, the geodesic distance from q to its furthest neighbor in C is at least 1-ε times the geodesic distance to its further neighbor in P. Thus the set C can be used for answering ε-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of P. The coreset can be constructed in O(1/(ε) (nlog(1/ε) + (n+m)log(n+m))) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Furthest-neighbor queries
  • polygons
  • geodesic distance
  • coreset

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References

  1. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL: https://doi.org/10.1145/1008731.1008736.
  2. Pankaj K. Agarwal, Sariel Har-Peled, and Hai Yu. Robust shape fitting via peeling and grating coresets. In Proc. 1th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 182-191, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109579.
  3. Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous. TSP in a Simple Polygon. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms (ESA 2022), volume 244 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:14, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.5.
  4. Boris Aronov, Steven Fortune, and Gordon T. Wilfong. The furthest-site geodesic Voronoi diagram. Discret. Comput. Geom., 9:217-255, 1993. URL: https://doi.org/10.1007/BF02189321.
  5. Franz Aurenhammer, Robert L. Scot Drysdale, and Hannes Krasser. Farthest line segment Voronoi diagrams. Inf. Process. Lett., 100(6):220-225, 2006. URL: https://doi.org/10.1016/j.ipl.2006.07.008.
  6. Franz Aurenhammer, Evanthia Papadopoulou, and Martin Suderland. Piecewise-linear farthest-site Voronoi diagrams. In 32nd International Symposium on Algorithms and Computation (ISAAC), volume 212 of LIPIcs, pages 30:1-30:11, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.30.
  7. Sergei Bespamyatnikh. Dynamic algorithms for approximate neighbor searching. In Proceedings of the 8th Canadian Conference on Computational Geometry, pages 252-257, 1996. URL: http://www.cccg.ca/proceedings/1996/cccg1996_0042.pdf.
  8. Otfried Cheong, Hazel Everett, Marc Glisse, Joachim Gudmundsson, Samuel Hornus, Sylvain Lazard, Mira Lee, and Hyeon-Suk Na. Farthest-polygon Voronoi diagrams. Comput. Geom., 44(4):234-247, 2011. URL: https://doi.org/10.1016/j.comgeo.2010.11.004.
  9. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications (3rd Edition). Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  10. Mark de Berg and Leonidas Theocharous. A coreset for approximate furthest-neighbor queries in a simple polygon. arXiv:2403.04513, 2024. URL: https://doi.org/10.48550/arXiv.2403.04513.
  11. Ashish Goel, Piotr Indyk, and Kasturi R. Varadarajan. Reductions among high dimensional proximity problems. In Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, pages 769-778, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365776.
  12. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci., 39(2):126-152, 1989. URL: https://doi.org/10.1016/0022-0000(89)90041-X.
  13. Piotr Indyk. Better algorithms for high-dimensional proximity problems via asymmetric embeddings. In Proc. 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 539-545, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644200.
  14. Kurt Mehlhorn, Stefan Meiser, and Ronald Rasch. Furthest site abstract Voronoi diagrams. Int. J. Comput. Geom. Appl., 11(6):583-616, 2001. URL: https://doi.org/10.1142/S0218195901000663.
  15. Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 633-701. North Holland / Elsevier, 2000. URL: https://doi.org/10.1016/b978-044482537-7/50016-4.
  16. Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The geodesic farthest-point Voronoi diagram in a simple polygon. Algorithmica, 82(5):1434-1473, 2020. URL: https://doi.org/10.1007/s00453-019-00651-z.
  17. Rasmus Pagh, Francesco Silvestri, Johan Sivertsen, and Matthew Skala. Approximate furthest neighbor with application to annulus query. Inf. Syst., 64:152-162, 2017. URL: https://doi.org/10.1016/j.is.2016.07.006.
  18. Michel Pocchiola and Gert Vegter. The visibility complex. Int. J. Comput. Geom. Appl., 6(3):279-308, 1996. URL: https://doi.org/10.1142/S0218195996000204.
  19. R. Pollack, M. Sharir, and G. Rote. Computing the geodesic center of a simple polygon. Discrete & Computational Geometry, 4(6):611-626, December 1989. URL: https://doi.org/10.1007/BF02187751.
  20. Raimund Seidel. On the number of faces in higher-dimensional Voronoi diagrams. In Proceedings of the Third Annual Symposium on Computational Geometry, pages 181-185, 1987. URL: https://doi.org/10.1145/41958.41977.
  21. Jack Snoeyink. Point location. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry (3rd edition), pages 767-785. Chapman and Hall/CRC, 2017. URL: https://doi.org/10.1201/9781315119601.
  22. 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.
  23. Godfried Toussaint. An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In Proceedings of EURASIP, Signal Processing III: Theories and Applications, Part 2, pages 853-856, 1986. Google Scholar
  24. Haitao Wang. An optimal deterministic algorithm for geodesic farthest-point voronoi diagrams in simple polygons. Discrete & Computational Geometry, 70(2):426-454, September 2023. URL: https://doi.org/10.1007/s00454-022-00424-6.
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