A modified Melzak procedure for computing node-weighted Steiner trees

Given a set of points in the Euclidean plane, the standard Steiner Problem asks for a network of minimum length connecting the points of β. A solution to this problem is called a Steiner Minimal Tree and may be computed exactly, using Melzak's compass-and-straightedge methods. A Steiner Tree often contains additional nodes, called Steiner points, not belonging to β. The node-weighted Steiner problem asks for the network connecting the points of β which minimizes the energy E = length + α(number of Steiner points), where α is a fixed nonnegative weight. Solutions to this problem, here called node-weighted Steiner trees, can have different geometries than those available to standard Steiner trees. This paper presents a modified Melzak procedure which computes the node-weighted Steiner tree for a given set β.