Optimal Voronoi diagram construction with n convex sites in three dimensions

Lee and Drysdale and Sharir have given suboptimal algorithms (n log2n) for computing the Voronoi diagrams of line-segments or discs in the plane. This paper adapts their methods to constructing the Voronoi diagram for disjoint convex sites in three dimensions.

Let C(n) be an upper bound on the number of features in any diagram involving n sites in three dimensions. Even for point sites, C(n) can be proportional to n2, so it is reasonable to assume that C(n)/n2 is monotonically nondecreasing. Under this assumption, our construction with n sites takes optimal time O(C(n)). Results of Aurenhammer's and Chazelle's show that if the sites are spherical then the diagram can be constructed in time O(n2), but C(n) has yet to be determined for more general classes of site. The sites are assumed to be compact and semi-algebraic of bounded algebraic complexity. For the sake of simplicity we assume that the sites are strictly convex and rounded (having no sharp corners nor edges). Assuming strict convexity simplifies the analysis, as does the roundedness assumption, though the latter is neither necessary nor restrictive.