The.Algorithm.Design.Manual.Springer-Verlag.1998

Voronoi Diagrams
McDonald's, the time it takes to get there depends upon where the major roads are. Efficient
algorithms are known for constructing Voronoi diagrams under a variety of different metrics, and
for curved or constrained objects.
● Power diagrams - These structures decompose space into regions of influence around the sites,
where the sites are no longer constrained to have all the same power. Imagine a map of the
listening range of a set of radio stations operating at a given frequency. The region of influence
around a station depends both on the power of its transmitter and the position of neighboring
transmitters.
● kth-order and furthest-site diagrams - The idea of decomposing space into regions sharing some
property can be taken beyond closest-point Voronoi diagrams. Any point within a single cell of
the kth-order Voronoi diagram shares the same set of k closest points in S. In furthest-site
diagrams, any point within a particular region shares the same furthest point in S. Point location
(see Section ) on these structures permits fast retrieval of the appropriate points.
Implementations: Steve Fortune is the author of a widely used 2D code for Voronoi diagrams and
Delaunay triangulations, written in C. It is based on his own sweepline algorithm [For87] for Voronoi
diagrams and is likely to be the right code to try first. It is available from Netlib (see Section ) at
http://netlib.bell-labs.com/netlib/voronoi/index.html.
LEDA (see Section ) provides an implementation of a randomized incremental construction algorithm
for planar Voronoi diagrams in C++.
Higher-dimensional and furthest-site Voronoi diagrams can be constructed as a special case of higherdimensional
convex hulls. Qhull [BDH97] appears to be the convex hull code of choice in three
dimensions and beyond. It is written in C, and it can also construct Delaunay triangulations and halfspace
intersections. Qhull has been widely used in scientific applications and has a well-maintained
home page at http://www.geom.umn.edu/software/qhull/.
The Stanford GraphBase (see Section ) contains an implementation of a randomized incremental
algorithm to construct Voronoi diagrams and Delaunay triangulations for use as a generator of planar
graph instances.
Algorithm 558 [Che80] of the Collected Algorithms of the ACM is a Fortran code for the multifacility
location problem. It is based on a network flow approach, instead of using Voronoi diagrams.
Interestingly, the network flow code is taken from Nijenhuis and Wilf (see Section ). See Section .
Notes: Voronoi diagrams were studied by Dirichlet in 1850 and are occasionally referred to as Dirichlet
tessellations. They are named after G. Voronoi, who discussed them in a 1908 paper. In mathematics,
concepts get named after the last person to discover them.
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