Constructing pairwise disjoint paths with few links

Let P be a simple polygon and let {( u 1 , u ′ 1 ), ( u 2 , u ′ 2 ),…,( u m , u ′ m )} be a set of m pairs of distinct vertices of P , where for every distinct i , j ≤ m , there exist pairwise disjoint (nonintersecting) paths connecting u i to u ′ i and u j to u ′ j . We wish to construct m pairwise disjoint paths in the interior of P connecting u i to u ′ i for i = 1, …, m , with a minimal total number of line segments. We give an approximation algorithm that constructs such a set of paths using O ( M ) line segments in O ( n log m + M log m ) time, where M is the number of line segments in the optimal solution and n is the size of the polygon.