Constructing pairwise disjoint paths with few links

Abstract

Let P be a simple polygon and let {(ui,u i ′ )} be m pairs of distinct vertices of P where for every distinct i, j ≤ m, there exist pairwise disjoint 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 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.