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While the standard unweighted Voronoi diagram in the plane has linearworst-case complexity, many of its natural generalizations do not. This paperconsiders two such previously studied generalizations, namely multiplicativeand semi Voronoi diagrams. These diagrams both have quadratic worst-casecomplexity, though here we show that their expected complexity is linear forcertain natural randomized inputs. Specifically, we argue that the expectedcomplexity is linear for: (1) semi Voronoi diagrams when the visible directionis randomly sampled, and (2) for multiplicative diagrams when either weightsare sampled from a constant-sized set, or the more challenging case whenweights are arbitrary but locations are sampled from a square.

Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams