E-mail: ijlee@cs.newcastle.edu.au, mark@geog.psu.edu

Ordinary Voronoi diagrams have gained popularity to model discrete point data sets, because they achieve the two issues above, namely, they uniquely model spatial proximity in points and algorithms and data structures exist for fast dynamic update using local spatial proximity. The information of spatial proximity is explicitly stored within the dual graph (the so-called Delaunay triangulation). So, a certain degree of "what-if" analysis is possible when points are updated using ordinary Voronoi diagrams. However, the ordinary Voronoi diagrams are informative and meaningful only when points have identical weight (relevance, interest or growth rate) or only when the nearest neighbour is the only relation of concern.

Indeed, a large number of domains demand "what-if" analysis of more
relations. Often, requiring (1) the *k*-th nearest neighbour (for
example, the *k*-th nearest hospital in case *k*-1 nearest hospitals
are closed or fully occupied), (2) the order-*k* nearest neighbour
(the *k* nearest customers for business marketing), (3) the ordered
order-*k* nearest neighbour (the ordered *k* nearest police stations
in case emergency occurs), (4) the farthest neighbour (the farthest area
from the pollutant) and (5) points having different weights (shopping centres
having different facilities result in different catchment areas). All these
five additional relations are modelled by generalised Voronoi diagrams:
(1) *k*-th nearest-point Voronoi diagram, (2) order-*k* Voronoi
diagram, (3) ordered order-*k* Voronoi diagram, (4) farthest Voronoi
diagram and (5) weighted Voronoi diagram, respectively. In this demonstration,
we present a GUI environment for "what-if" analysis when points have all
the different meanings (distance concepts and weights) just mentioned.

The implemented application utilises our innovative common data structure, which supports jointly the ordinary Voronoi diagram and the five generalisations. This data structure holds spatial adjacency information with respect to generalised Voronoi vertices. That is, it is possible to derive all generalised Voronoi diagrams from this single data structure without storing the variants explicitly. In addition, spatial proximity information is easily retrieved, including generalised Voronoi regions and neighbours. Further, we demonstrate the ease by which our application facilitates compare and contrast amongst all the Voronoi diagrams within a common data set. Alternation from one diagram to another is also easily performed. In summary, our product enables users to effectively construct and interact with much more comprehensive "what-if" scenarios that could involve applications analysing complex cost functions or involve risk factors.

Download the application: voronoi.exe