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. Assume that each point is chosen independently and identically from the subset of points of the same color, we propose an O(n 2logn) time algorithm to compute the expected area of convex hulls of the color spanning sets and an O(n 2logn) 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(nlogn) (resp. O(n 2)) time algorithm. We also propose an approximation algorithm to compute the expected diameter of the color spanning sets. © 2013 Springer-Verlag Berlin Heidelberg.