On Approximations for Constructing 1-Line Minimum Rectilinear Steiner Trees in the Euclidean Plane $\mathbb {R}^2$

Abstract

In this paper, we consider the 1-line minimum rectilinear Steiner tree (1L-MRST) problem, which is defined as follows. Given n points in the Euclidean plane $\mathbb {R}^2$, we are asked to find the location of a line l and a Steiner tree T(l), which consists of vertical and horizontal line segments plus the line l, to interconnect these n points and at least one point on the line l, the objective is to minimize total weight of T(l), i.e., $\min \{\sum _{uv\in T(l)} w(u,v)$ | T(l) is a Steiner tree as mentioned-above$\}$, where weight $w(u,v)=0$ if two endpoints u, v of an edge $uv \in T(l)$ is located on the line l and weight w(u, v) as the rectilinear distance between u and v otherwise. Given a line l as an input, we refer to this problem as the 1-line-fixed minimum rectilinear Steiner tree (1LF-MRST) problem; In addition, when Steiner points of T(l) are all located on the line l, we refer to this problem problem as the constrained minimum rectilinear Steiner tree (CMRST) problem.

We obtain three main results as follows. (1) We design an exact algorithm in time $O(n\log n)$ to solve the CMRST problem; (2) We show that the same algorithm in (1) is a 1.5-approximation algorithm to solve the 1LF-MRST problem; (3) Using a combination of the algorithm in (1) for many times and a key lemma proved by using some techniques of computational geometry, we provide a 1.5-approximation algorithm in time $O(n^3\log n)$ to solve the 1L-MRST problem.

This paper is supported by Project of the National Natural Science Foundation of China [Nos. 11861075, 11801498], Project for Innovation Team (Cultivation) of Yunnan Province, Joint Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No. 2018FY001014] and IRTSTYN. In addition, J.R. Lichen is also supported by Project of Doctorial Fellow Award of Yunnan Province [No. 2018010514].