# Parallel algorithms for finding Steiner forests in planar graphs

## Abstract

Given an unweighted planar graph *G* together with nets of terminals, our problem is to find a Steiner forest, i.e., vertex-disjoint trees, each of which interconnects all the terminals of a net. This paper presents four parallel algorithms for the Steiner forest problem and a related one. The first algorithm solves the problem for the case all the terminals are located on the outer boundary of *G* in *O*(log^{2}*n*) time using *O*(*n*^{3}/log *n*) processors on a CREW PRAM, where *n* is the number of vertices in *G*. The second algorithm solves the problem for the case all terminals of each net lie on one of a fixed number of face boundaries in poly-log time using a polynomial number of processors. The third solves the problem for the case all terminals lie on two face boundaries. The fourth finds a maximum number of internally disjoint paths between two specified vertices in planar graphs. Both the third and fourth run either in *O*(log^{2}*n*) time using *O*(*n*^{6}/log *n*) processors or in (log^{3}*n*) time using *O*(*n*^{3}/log *n*) processors.

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