The Steiner tree problem is a well known network optimization problem which asks for a connected minimum network (called a Steiner minimum tree) spanning a given point set N. In the original Steiner tree problem the given points lie in the Euclidean plane or space, and the problem has many variants in different applications now. Recently a new type of Steiner minimum tree, probability Steiner minimum tree, is introduced by the authors in the study of phylogenies. A Steiner tree is a probability Steiner tree if all points in the tree are probability vectors in a vector space. The points in a Steiner minimum tree (or a probability Steiner tree) that are not in the given point set are called Steiner points (or probability Steiner points respectively). In this paper we investigate the properties of Steiner points and probability Steiner points, and derive the formulae for computing Steiner points and probability Steiner points in ℓ1- and ℓ2-metric spaces. Moreover, we show by an example that the length of a probability Steiner tree on 3 points and the probability Steiner point in the tree are smooth functions with respect to p in d-space.
A flow-dependent quadratic steiner tree problem in the Euclidean plane
