Shortest noncrossing paths in plane graphs

Let G be an undirected plane graph with nonnegative edge length, and letk terminal pairs lie on two specified face boundaries. This paper presents an algorithm for findingk noncrossing paths inG, each connecting a terminal pair, and whose total length is minimum. Noncrossing paths may share common vertices or edges but do not cross each other in the plane. The algorithm runs in timeO(n logn) wheren is the number of vertices inG andk is an arbitrary integer.