In this paper, motivated by many practical applications, we address the 1-line minimum rectilinear Steiner tree (1L-MRStT) problem, which is a variation of the Euclidean minimum rectilinear Steiner tree problem. More specifically, given n points in the Euclidean plane \({\mathbb {R}}^2\), it is asked to find the location of a line l and a Steiner tree T(l), consisting only of vertical and horizontal line segments plus several successive segments located on this line l, to interconnect these n points and at least one point located on the line l, the objective is to minimize total weight of this Steiner tree T(l), i.e., \(\min \{\sum _{uv\in T(l)} w(u,v)\) | T(l) is a Steiner tree mentioned-above\(\}\), where we define a weight \(w(u,v)=0\) if the two endpoints u and v of that edge \(uv \in T(l)\) is located on the line l and otherwise we define a weight w(u, v) as the rectilinear distance between the two endpoints u and v of that edge \(uv \in T(l)\). Given a line l as an input in \({\mathbb {R}}^2\), we denote this problem as the 1-line-fixed minimum rectilinear Steiner tree (1LF-MRStT) problem; Furthermore, when the Steiner points of T(l) are all located on the fixed line l, we recall this problem as the 1-line-fixed-constrained minimum rectilinear Steiner tree (1LFC-MRStT) problem. We provide three following main contributions. (1) We design an algorithm \({{\mathcal {A}}}_{C}\) to optimally solve the 1LFC-MRStT problem, where the algorithm \({{\mathcal {A}}}_{C}\) runs in time \(O(n\log n)\); (2) We prove that this algorithm \({{\mathcal {A}}}_{C}\) is a 1.5-approximation algorithm to solve the 1LF-MRStT problem; (3) Combining the algorithm \({{\mathcal {A}}}_{C}\) for many times and a key lemma proved by some techniques of computational geometry, we present a 1.5-approximation algorithm to solve the 1L-MRStT problem, where this algorithm runs in time \(O(n^3\log n)\), and we finally provide another approximation algorithm to solve a special version of the 1L-MRStT problem, where that new algorithm runs in lower time \(O(n^2\log n)\).

在本文中，受许多实际应用的启发，我们解决了 1 行最小直线 Steiner 树 (1L-MRStT) 问题，它是欧几里德最小直线 Steiner 树问题的变体。更具体地说，给定欧几里得平面\({\mathbb {R}}^2\)中的n个点，要求找到一条线l和一个 Steiner 树T ( l )的位置，该 树仅由垂直和水平组成线段加上位于这条线l上的几个连续线段 ，将这n个点和位于线l上的至少一个点连接起来，目标是最小化这个 Steiner 树T 的总权重( l )，即，\(\min \{\sum _{uv\in T(l)} w(u,v)\) | Ť（升）是斯坦纳树提到-以上\（\} \） ，在这里我们定义一个重量\（W（U，V）= 0 \）如果两个端点ù和v该边缘的\（UV \在T(l)\)位于线l 上，否则我们将权重w ( u ,  v )定义为该边\(uv \in T(l)\)的两个端点u和v之间的直线距离。给定一条线l作为\({\mathbb {R}}^2\) 的输入，我们将此问题表示为 1-line-fixed minimum rectilinear Steiner tree (1LF-MRStT) 问题；此外，当T ( l )的 Steiner 点都位于固定线l 上时，我们将这个问题称为 1 线固定约束最小直线 Steiner 树 (1LFC-MRStT) 问题。我们提供以下三个主要贡献。(1) 我们设计了一个算法\({{\mathcal {A}}}}_{C}\)来最优解决 1LFC-MRStT 问题，其中算法\({{\mathcal {A}}}}_{C }\)及时运行\(O(n\log n)\) ; (2) 我们证明这个算法\({{\mathcal {A}}}_{C}\)是解决 1LF-MRStT 问题的 1.5 近似算法；(3)多次结合算法\({{\mathcal {A}}}_{C}\)和计算几何的一些技术证明的一个关键引理，我们提出了一个1.5-近似算法来解决1L- MRStT 问题，该算法在时间上运行\(O(n^3\log n)\)，我们最终提供了另一种近似算法来解决 1L-MRStT 问题的特殊版本，其中新算法在更短的时间内运行\(O(n^2\log n)\)。