Algorithms for Finding Noncrossing Steiner Forests in Plane Graphs

Abstract

Let G = (V,E) be a plane graph with nonnegative edge lengths, and let N be a family of k vertex sets N ₁,N ₂,...,N _k $\subseteq$ V, called nets. Then a noncrossing Steiner forest for N in G is a set T of k trees T ₁, T ₂,...,T _k in G such that each tree T _i ∈ T connects all vertices in N _i , any two trees in T do not cross each other, and the sum of edge lengths of all trees is minimum. In this paper we give an algorithm to find a noncrossing Steiner forest in a plane graph G for the case where all vertices in nets lie on two of the face boundaries of G. The algorithm takes time O(n log n) if G has n vertices.