The Overlay of Minimization Diagrams in a Randomized Incremental Construction

In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in ℝ^d, for d≥2, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n^⌊d/2⌋log n), for d odd, and O(n^⌊d/2⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d=3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.