Approximation Algorithms for Solving the 1-Line Minimum Steiner Tree of Line Segments Problem

We address the 1-line minimum Steiner tree of line segments (1L-MStT-LS) problem. Specifically, given a set S of n disjoint line segments in R-2, we are asked to find the location of a line l and a set E-l of necessary line segments (i.e., edges) such that a graph consisting of all line segments in S. E-l plus this line l, denoted by T-l = (S, l, E-l), becomes a Steiner tree, the objective is to minimize total length of edges in E-l among all such Steiner trees. Similarly, we are asked to find a set E-0 of necessary edges such that a graph consisting of all line segments in S boolean OR E-0, denoted by T-S = (S, E-0), becomes a Steiner tree, the objective is to minimize total length of edges in E-0 among all such Steiner trees, we refer to this new problem as the minimum Steiner tree of line segments (MStT-LS) problem. In addition, when two endpoints of each edge in E-0 need to be located on two different line segments in S, respectively, we refer to that problem as the minimum spanning tree of line segments (MST-LS) problem. We obtain three main results: (1) Using technique of Voronoi diagram of line segments, we design an exact algorithm in time O(n log n) to solve the MST-LS problem; (2) we show that the algorithm designed in (1) is a 1.214-approximation algorithm to solve the MStT-LS problem; (3) using the combination of the algorithm designed in (1) as a subroutine for many times, a technique of finding linear facility location and a key lemma proved by techniques of computational geometry, we present a 1.214-approximation algorithm in time O(n(3) log n) to solve the 1L-MStT-LS problem.