Finding a Noncrossing Steiner Forest in Plane Graphs Under a 2-Face Condition

Abstract

Let G = (V,E) be a plane graph with nonnegative edge weights, and let $\mathcal{N}$ be a family of k vertex sets $N_1 ,N_2 ,...,N_k \subseteq V$, called nets. Then a noncrossing Steiner forest for $\mathcal{N}$ in G is a set $\mathcal{T}$ of k trees $T_1 ,T_2 ,...,T_k$ in G such that each tree $T_i \in \mathcal{T}$ connects all vertices, called terminals, in net N _i, any two trees in $\mathcal{T}$ do not cross each other, and the sum of edge weights 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 terminals in nets lie on any two of the face boundaries of G. The algorithm takes time $O\left( {n\log n} \right)$ if G has n vertices and each net contains a bounded number of terminals.