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Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams

Author Luis Barba

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Luis Barba
  • Department of Computer Science, ETH Zürich, Switzerland

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Luis Barba. Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


Let P be a simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. Given a set S of m sites being a subset of the vertices of P, we present the first randomized algorithm to compute the geodesic farthest-point Voronoi diagram of S in P running in expected O(n + m) time. That is, a partition of P into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. This algorithm can be extended to run in expected O(n + m log m) time when S is an arbitrary set of m sites contained in P.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Geodesic distance
  • simple polygons
  • farthest-point Voronoi diagram


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