Approximate Voronoi Diagrams

A Voronoi diagram of a point set P ⊆ IR is a partition of space into regions, such that the cell of p ∈ P , is V (p, P ) = { x ∣∣∣‖xp‖ ≤‖xp′‖ for all p′ ∈ P }. Vornoi diagrams are a powerful tool and have numerous applications [Aur91]. One problem with Vornoi diagrams is that their descriptive complexity is O(ndd/2e) in IR, in the worst case. See Figure 1 for an exmaple in three dimensions. It is a natrual question to ask, whether one can reduce the complexity to linear (or near linear) by allow some approximation.