2026-07-23T20:03:00Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8771 2024-03-06T10:42:37Z ddc:004 open_access

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon Liu, Chih-Hung Simple polygons Voronoi diagrams Geodesic distance 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Chih-Hung Liu 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.58 urn:nbn:de:0030-drops-87717 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.58 eng https://creativecommons.org/licenses/by/3.0/legalcode