Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams

While the standard unweighted Voronoi diagram in the plane has linear worst-case complexity, many of its natural generalizations do not. This paper considers two such previously studied generalizations, namely multiplicative and semi Voronoi diagrams. These diagrams both have quadratic worst-case complexity, though here we show that their expected complexity is linear for certain natural randomized inputs. Specifically, we argue that the expected complexity is linear for: (1) semi Voronoi diagrams when the visible direction is randomly sampled, and (2) for multiplicative diagrams when weights are sampled from a constant-sized set. For the more challenging case, when weights are arbitrary but fixed, and the locations are sampled from the unit square, we argue the expected complexity of the multiplicative diagram within the unit square is linear.