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Abstract:
Abstract Voronoi diagrams were introduced by R.~Klein
as an axiomatic basis of
Voronoi diagrams.
We show how to construct abstract Voronoi diagrams in time
$O(n\log n)$ by a randomized algorithm,
which is based on Clarkson and Shor's randomized
incremental construction technique.
The new algorithm has the following advantages over
previous algorithms:
\begin{itemize}
\item
It can handle a much wider class of abstract Voronoi
diagrams than the algorithms presented in [Kle89b, MMO91].
\item
It can be adapted to a concrete kind of Voronoi diagram by
providing a single basic operation, namely the
construction of a Voronoi diagram of five sites.
Moreover, all geometric decisions are confined to the
basic operation, and using this operation, abstract
Voronoi diagrams can be constructed in a purely
combinatorial manner.
