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

Authors Eunjin Oh, Hee-Kap Ahn

Thumbnail PDF


  • Filesize: 0.61 MB
  • 15 pages

Document Identifiers

Author Details

Eunjin Oh
Hee-Kap Ahn

Cite AsGet BibTex

Eunjin Oh and Hee-Kap Ahn. Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Given a set of sites in a simple polygon, a geodesic Voronoi diagram 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 < n/polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m < n/polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.
  • Simple polygons
  • Voronoi diagrams
  • geodesic distance


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. 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. Google Scholar
  2. Boris Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica, 4(1):109-140, 1989. Google Scholar
  3. Boris Aronov, Steven Fortune, and Gordon Wilfong. The furthest-site geodesic Voronoi diagram. Discrete &Computational Geometry, 9(1):217-255, 1993. Google Scholar
  4. Jon Louis Bentley and James B. Saxe. Decomposable searching problems 1: Static-to-dynamic transformations. Journal of Algorithms, 1(4):297-396, 1980. Google Scholar
  5. Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994. Google Scholar
  6. Herbert Edelsbrunner and Ernst Peter Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9(1):66-104, 1990. Google Scholar
  7. Steven Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2(1):153-174, 1987. Google Scholar
  8. Leonidas Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1):209-233, 1987. Google Scholar
  9. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989. Google Scholar
  10. John Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991. Google Scholar
  11. Chih-Hung Liu and D. T. Lee. 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), pages 1633-1645, 2013. Google Scholar
  12. Nimrod Megiddo. Linear-time algorithms for linear programming in ℝ³ and related problems. SIAM Journal on Computing, 12(4):759-776, 1983. Google Scholar
  13. Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000. Google Scholar
  14. Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The farthest-point geodesic voronoi diagram of points on the boundary of a simple polygon. In Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016), pages 56:1-56:15, 2016. Google Scholar
  15. Evanthia Papadopoulou and D. T. Lee. A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica, 1998(4):319-352, 1998. Google Scholar
  16. Richard Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discrete &Computational Geometry, 4(6):611-626, 1989. Google Scholar
  17. Maksym Zavershynskyi and Evanthia Papadopoulou. A sweepline algorithm for higher order voronoi diagrams. In Proceedings of the 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2013), pages 16-22, 2013. Google Scholar