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Given a set S of n sites in the plane, the Voronoi diagram is a partition of the plane into regions, each region consisting of the set of points closer to a particular site in S than to any other site in S [PS85]. The literature on Voronoi diagrams and various extensions to Voronoi diagrams is extensive and has been recently surveyed by Aurenhammer [Aur88]. One extension is the mtdtiplicatively weighted Voronoi diagram in which the distance between a site p and a point X is de (P, x)/W(P), where de (p, X) is the Euclidean distance between p and X. The multiplicatively weighted diagram has size @(n2 ) and @(n2 ) time algorithms to compute it are known [AE84, Aur87]. There are applications of multiplicative weighted diagrams in many fields, including geography and economics [GHJ67, HL79, HJ68]. The standard Voronoi diagram and the additively weighted Voronoi diagram are widely used to model crystal growth [M HM80, GB78]. Modeling crystal growth using the standard Voronoi diagram or the addltively weighted Voronoi diagram assumes all crystals have the same constant growth rate. For many physical systems, this may not be a valid assumption [Guy72]. Frost and Thompson [FT87, FT88] use many variants of the Voronoi diagram to simulate crystal growth on thin films at many different growth rates. When each crystal hss a constant growth rate and the constant is allowed to be different for each crystal, they simulate the crystal growth by using the multiplicatively weighted Voronoi diagram. They note that this simulation is an approximation since the multiplicatively weighted Voronoi diagram produces regions that are not connected. Crystals growing from sites will always pr~ duce connected regions. In this paper, we propose a new model of the multiplicatively weighted Voronoi diagram that can be used to accurately simulate multiplicatively weighted crystal growth. In this new model, the distance from a site to a point in its region is measured along a shortest path lying entirely within the region. In the multiplicatively weighted crystal growth Voronoi diagram, a growing crystal (or region) may “wrap around” another site’s region. When a

Multiplicatively weighted crystal growth Voronoi diagrams (extended abstract)