CHIH-HUNG LIU2022-11-112022-11-11201901784617https://www.scopus.com/inward/record.uri?eid=2-s2.0-85073804375&doi=10.1007%2fs00453-019-00624-2&partnerID=40&md5=5881b1263d95877d1f84c819575c3552https://scholars.lib.ntu.edu.tw/handle/123456789/624637The 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 Ω(n+ mlog 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+ mlog mlog 2n) time, which are optimal for m= Ω(n) and m=O(nlog3n), respectively. In this paper, we give a construction algorithm with O(n+ m(log m+ log 2n)) 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+ mlog 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. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.Computational geometry; Geodesic distance; Simple polygons; Voronoi diagrams[SDGs]SDG11Geodesy; Graphic methods; Construction algorithms; Construction time; Geodesic distances; Geodesic voronoi diagram; Optimal algorithm; Simple polygon; State of the art; Voronoi diagrams; Computational geometryjournal article10.1007/s00453-019-00624-22-s2.0-85073804375