  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.
