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A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

Author Chih-Hung Liu

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Chih-Hung Liu

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Chih-Hung Liu. A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 58:1-58:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.
  • Simple polygons
  • Voronoi diagrams
  • Geodesic distance


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