Area and Perimeter of the Convex Hull of Stochastic Points

Given a set P of n points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset S of P . The random subset S is formed by drawing each point p of P independently with a given rational probability πp. For both measures of the convex hull, we show that it is #P-hard to compute the probability that the measure is at least a given bound w. For ε ∈ (0, 1), we provide an algorithm that runs in O(n/ε) time and returns a value that is between the probability that the area is at least w, and the probability that the area is at least (1 − ε)w. For the perimeter, we show a similar algorithm running in O(n/ε) time. Finally, given ε, δ ∈ (0, 1) and for any measure, we show an O(n log n+(n/ε) log(1/δ))-time Monte Carlo algorithm that returns a value that, with probability of success at least 1− δ, differs at most ε from the probability that the measure is at least w.