Faster algorithms for the minimum red-blue-purple spanning graph problem

Consider a set of n points in the plane, each one of which is colored either red, blue, or purple. A red-blue-purple spanning graph (RBP spanning graph) is a graph whose vertices are the points and whose edges connect the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. The minimum RBP spanning graph problem is to find an RBP spanning graph with minimum total edge length. First we consider this problem for the case when the points are located on a circle. We present an algorithm that solves this problem in O(n2) time, improving upon the previous algorithm by a factor of Θ(n). Also, for the general case we present an algorithm that runs in O(n5) time, improving upon the previous algorithm by a factor of Θ(n)