Constructing Pairwise Disjoint Paths with

Let P be a simple polygon and let f(ui; u 0 i)g be m pairs of distinct vertices of P where for every distinct i; j m, there exist pairwise disjoint paths connecting ui to u 0 i and uj to u 0 j. We wish to construct m pairwise disjoint paths in the interior of P connecting ui to u 0 i for i = 1; : : : ; m, with minimal total number of line segments. We give an approximation algorithm which in O(n log m + M log m) time constructs such a set of paths using O(M) line segments where M is the number of line segments in the optimal solution.