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Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

Authors Mincheol Kim, Chanyang Seo, Taehoon Ahn, Hee-Kap Ahn



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Author Details

Mincheol Kim
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Chanyang Seo
  • Graduate School of Artificial Intelligence, Pohang University of Science and Technology, South Korea
Taehoon Ahn
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Hee-Kap Ahn
  • Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea

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Mincheol Kim, Chanyang Seo, Taehoon Ahn, and Hee-Kap Ahn. Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 51:1-51:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.51

Abstract

We present an algorithm to compute the geodesic L₁ farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes O(nm+n log n + mlog m) construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in O(log(n+m)) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geodesic distance
  • L₁ metric
  • farthest-point Voronoi diagram

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References

  1. A. Aggarwal, L.J. Guibas, J. Saxe, and P.W. Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete & Computational Geometry, 4(6):591-604, 1989. Google Scholar
  2. H. Alt, O. Cheong, and A. Vigneron. The Voronoi diagram of curved objects. Discrete & Computational Geometry, 34(3):439-453, 2005. Google Scholar
  3. B. Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica, 4(1):109-140, 1989. Google Scholar
  4. B. Aronov, S. Fortune, and G. Wilfong. The furthest-site geodesic Voronoi diagram. Discrete & Computational Geometry, 9(3):217-255, 1993. Google Scholar
  5. S.W. Bae and K.-Y. Chwa. The geodesic farthest-site Voronoi diagram in a polygonal domain with holes. In Proceedings of the 25th Annual Symposium on Computational Geometry (SoCG), pages 198-207, 2009. Google Scholar
  6. B. Ben-Moshe, B.K. Bhattacharya, and Q. Shi. Farthest neighbor Voronoi diagram in the presence of rectangular obstacles. In Proceedings of the 13th Canadian Conference on Computational Geometry (CCCG), pages 243-246, 2005. Google Scholar
  7. B. Ben-Moshe, M.J. Katz, and J.S.B. Mitchell. Farthest neighbors and center points in the presence of rectangular obstacles. In Proceedings of the 17th Annual Symposium on Computational Geometry (SoCG), pages 164-171, 2001. Google Scholar
  8. O. Cheong, H. Everett, M. Glisse, J. Gudmundsson, S. Hornus, S. Lazard, M. Lee, and H.-S. Na. Farthest-polygon Voronoi diagrams. Computational Geometry, 44(4):234-247, 2011. Google Scholar
  9. L.P. Chew and R.L. Dyrsdale III. Voronoi diagrams based on convex distance functions. In Proceedings of the 1st annual symposium on Computational geometry (SoCG), pages 235-244, 1985. Google Scholar
  10. J. Choi, C.-S. Shin, and S.K. Kim. Computing weighted rectilinear median and center set in the presence of obstacles. In International Symposium on Algorithms and Computation, pages 30-40. Springer, 1998. Google Scholar
  11. J. Choi and C. Yap. Monotonicity of rectilinear geodesics in d-space. In Proceedings of the 12th Annual Symposium on Computational Geometry (SoCG), pages 339-348, 1996. Google Scholar
  12. P.J. De Rezende, D.-T. Lee, and Y.-F. Wu. Rectilinear shortest paths with rectangular barriers. In Proceedings of the 1st Annual Symposium on Computational Geometry (SoCG), pages 204-213, 1985. Google Scholar
  13. H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements. Discrete & Computational Geometry, 1(1):25-44, 1986. Google Scholar
  14. S. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2(1):153-174, 1987. Google Scholar
  15. Y. Giora and H. Kaplan. Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Transactions on Algorithms, 5(3):28:1-51, 2009. Google Scholar
  16. J. Hershberger and S. Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. Google Scholar
  17. R. Klein. Abstract Voronoi diagrams and their applications. In Proceedings of the 4th International Workshop on Computational Geometry (EuroCG), pages 148-157. Springer, 1988. Google Scholar
  18. D.-T. Lee. Two-dimensional Voronoi diagrams in the L_p-metric. Journal of the ACM, 27(4):604-618, 1980. Google Scholar
  19. E. Oh. Optimal algorithm for geodesic nearest-point Voronoi diagrams in simple polygons. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 391-409, 2019. Google Scholar
  20. E. Oh and H.-K. Ahn. Voronoi diagrams for a moderate-sized point-set in a simple polygon. Discrete & Computational Geometry, 63(2):418-454, 2020. Google Scholar
  21. E. Oh, L. Barba, and H.-K. Ahn. The geodesic farthest-point Voronoi diagram in a simple polygon. Algorithmica, 82(5):1434-1473, 2020. Google Scholar
  22. E. Papadopoulou and S.K. Dey. On the farthest line-segment Voronoi diagram. International Journal of Computational Geometry & Applications, 23(06):443-459, 2013. Google Scholar
  23. E. Papadopoulou and D.T. Lee. The L_∞oronoi diagram of segments and VLSI applications. International Journal of Computational Geometry & Applications, 11(05):503-528, 2001. Google Scholar
  24. M.I. Shamos and D. Hoey. Closest-point problems. In Proceedings of the 16th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 151-162, 1975. Google Scholar
  25. H. Wang. An optimal deterministic algorithm for geodesic farthest-point Voronoi diagrams in simple polygons. In Proceedings of the 37th International Symposium on Computational Geometry (SoCG), pages 59:1-59:15, 2021. Google Scholar
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