# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Thursday, April 19, 2018, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Chih-Hung Liu

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

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^2 n ) time, which are optimal for m = Omega(n) and m = O( n / log^3 n ), 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. This work will appear in the International Symposium on Computational Geometry, 2018.

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