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

Let G = (V, E) be a plane graph with nonnegative edge weights, and letN be a family of k vertex sets N1, N2, . . . , Nk ⊆ V , called nets. Then a noncrossing Steiner forest forN in G is a set T of k trees T1, T2, . . . , Tk in G such that each tree Ti ∈ T connects all vertices, called terminals, in net Ni , any two trees in 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(n log n) if G has n vertices and each net contains a bounded number of terminals.