text: Given a set @math of @math 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 @math of @math . The random subset @math is formed by drawing each point @math of @math independently with a given rational probability @math . 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 @math . For @math , we provide an algorithm that runs in @math time and returns a value that is between the probability that the area is at least @math , and the probability that the area is at least @math . For the perimeter, we show a similar algorithm running in @math time. Finally, given @math and for any measure, we show an @math -time Monte Carlo algorithm that returns a value that, with probability of success at least @math , differs at most @math from the probability that the measure is at least @math .
  @cite_6: We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points s1,s_2,...,sn in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability pi. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. We show that the stochastic MST problem is SPHARD for any dimension d ≥ 2. We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. For d=2, we present two deterministic approximation algorithms: an O(n4)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic.
  @cite_8: Given a set of @math n points, each is painted by one of the @math k given colors, we want to choose @math k points with distinct colors to form a color spanning set. For each color spanning set, we can construct the convex hull and the smallest axis-aligned enclosing rectangle, etc. Assuming that each point is chosen independently and identically from the subset of points of the same color, we propose an @math O ( n 2 ) time algorithm to compute the expected area of convex hulls of the color spanning sets and an @math O ( n 2 ) time algorithm to compute the expected perimeter of convex hulls of the color spanning sets. For the expected perimeter (resp. area) of the smallest perimeter (resp. area) axis-aligned enclosing rectangles of the color spanning sets, we present an @math O ( n log n ) (resp. @math O ( n 2 ) ) time algorithm. We also propose a simple approximation algorithm to compute the expected diameter of the color spanning sets. For the expected distance of the closest pair, we show that it is @math # P-complete to compute and there exists no polynomial time @math 2 n 1 - ? approximation algorithm to compute the probability that the closest pair distance of all color spanning sets equals to a given value @math d unless @math P = N P , even in one dimension and when each color paints two points.
summary: More recently, in 2014, @cite_8 considered a set of @math points in the plane colored with @math colors, and studied, among other computation problems, the computation of the expected area or perimeter of the convex hull of a random sample of the points. Such random samples are obtained by picking for each color a point of that color uniformly at random. They proved that both expectations can be computed in @math time. We note that their arguments can be used to compute both @math and @math , each one in @math time. In the case of the expected perimeter, similar arguments were discussed by @cite_6 .