# Algorithms for Finding Noncrossing Steiner Forests in Plane Graphs

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## 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.

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© Springer-Verlag Berlin Heidelberg 1999