We address the 1-line Steiner tree problem with minimum number of Steiner points. Given a line l, a point set P of n terminals in \({\mathbb {R}}^2\) and a positive constant K, we are asked to find a Steiner tree \(T_{l}\) to interconnect the line l and the n terminals such that the Euclidean length of each edge in \(T_{l}\) is no more than the given positive constant K except those connecting two points on the line l, the objective is to minimize total number of the Steiner points in \(T_{l}\), i.e. \(\min _{T_{l}}\{|S_{out}|+|S_{on}|\}\), where \(|S_{out}|\) and \(|S_{on}|\) are the number of Steiner points located outside of the line l and on this line l, respectively. We design a 4-approximation algorithm with time complexity of \(O(n^3)\) for the 1-line Steiner tree problem with minimum number of Steiner points.

我们解决具有最小数量 Steiner 点的 1 行 Steiner 树问题。给定一条线l 、一个由 \({\mathbb {R}}^2\)中n 个端点组成的点集P和一个正常数K，我们需要找到一个 Steiner 树\(T_{l}\)来互连直线l和n 个端点，使得\(T_{l}\)中每条边的欧几里得长度不超过给定的正常数K，除了连接直线l上两点的点之外，目标是最小化总数\(T_{l}\)中斯坦纳点的个数，即\(\min _{T_{l}}\{|S_{out}|+|S_{on}|\}\)，其中\(|S_{out}|\)和\(|S_{on} |\)分别是位于线l之外和这条线l上的 Steiner 点的数量。我们针对具有最少 Steiner 点数的 1 行 Steiner 树问题设计了时间复杂度为O(n^3)的 4 近似算法。