  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 either weights
are sampled from a constant-sized set, or the more challenging case when
weights are arbitrary but locations are sampled from a square.
