Expected computations on color spanning sets

Abstract

Given a set of $n$ points, each is painted by one of the $k$ given colors, we want to choose $k$ 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 $O(n^2)$ time algorithm to compute the expected area of convex hulls of the color spanning sets and an $O(n^2)$ 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 $O(n\log n)$ (resp. $O(n^2)$) 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 $\#$P-complete to compute and there exists no polynomial time $2^{n^{1-\varepsilon }}$ approximation algorithm to compute the probability that the closest pair distance of all color spanning sets equals to a given value $d$ unless $P=NP$, even in one dimension and when each color paints two points.