Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Abstract

Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for \(m \le n/{\text {polylog}}n\). Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for \(m \le n/{\text {polylog}}n\). This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Ahn, H.-K., Barba, L., Bose, P., De Carufel, J.-L., Korman, M., Oh, E.: A linear-time algorithm for the geodesic center of a simple polygon. Discrete Comput. Geom. 56(4), 836–859 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aronov, B.: On the geodesic Voronoĭ diagram of point sites in a simple polygon. Algorithmica 4(1), 109–140 (1989)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic Voronoĭ diagram. Discrete Comput. Geom. 9(1), 217–255 (1993)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bentley, J.L., Saxe, J.B.: Decomposable searching problems I: static-to-dynamic transformations. J. Algorithms 1(4), 297–396 (1980)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chazelle, B.: The Discrepancy Method. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  6. 6.

    Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Hershberger, J., Sharir, M., Snoeyink, J.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12(1), 54–68 (1994)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fortune, S.: A sweepline algorithm for Voronoĭ diagrams. Algorithmica 2, 153–174 (1987)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci. 39(2), 126–152 (1989)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hershberger, J.: A new data structure for shortest path queries in a simple polygon. Inform. Process. Lett. 38(5), 231–235 (1991)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Liu, C.-H., Lee, D.T.: Higher-order geodesic Voronoi diagrams in a polygonal domain with holes. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pp. 1633–1645. SIAM, Philadelphia (2013)

  12. 12.

    Megiddo, N.: Linear-time algorithms for linear programming in ${{\mathbb{R}}}^3$ and related problems. SIAM J. Comput. 12(4), 759–776 (1983)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. North-Holland, Amsterdam (2000)

    Google Scholar 

  14. 14.

    Oh, E., Barba, L, Ahn, H.-K.: The farthest-point geodesic voronoi diagram of points on the boundary of a simple polygon. In: Fekete, S., Lubiw, A. (eds.) Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016). LIPIcs. Leibniz International Proceedings in Informatics, vol. 51, pp. 56:1–56:15. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2016)

  15. 15.

    Papadopoulou, E., Lee, D.T.: A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica 20(4), 319–352 (1998)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(6), 611–626 (1989)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zavershynskyi, M., Papadopoulou, E.: A sweepline algorithm for higher order voronoi diagrams. In: Proceedings of the 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2013), pp. 16–22. IEEE (2013)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hee-Kap Ahn.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP-2017-0-00905) supervised by the IITP (Institute for Information & communications Technology Promotion).

Editor in Charge: Kenneth Clarkson

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Oh, E., Ahn, HK. Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon. Discrete Comput Geom 63, 418–454 (2020). https://doi.org/10.1007/s00454-019-00063-4

Download citation

Keywords

  • Voronoi diagrams
  • Geodesic distance
  • Simple Polygons

Mathematics Subject Classification

  • 65D18