Probabilistic analysis for combinatorial functions of moving points

Abstract

We initiate a probabilistic study of configuration functions of moving points. In our probabilistic model, a particle is given an initial position and a velocity drawn independently at random from the same distribution D. We show that if n particles are drawn independently at random from the uniform distribution on the square, their convex hull undergoes Θ(log2 n) combinatorial changes in expectation, their Voronoi diagram undergoes Θ(n3/2) combinatorial changes, and their closest pair undergoes Θ(n) combinatorial changes.