On the minimum number of Steiner points of constrained 1-line-fixed Steiner tree in the Euclidean plane ${\mathbb {R}}^2$

Abstract

In this paper, we address the problem of minimum number of Steiner points of constrained 1-line-fixed Steiner tree (abbreviated to the MNSP-C1LF-ST problem), which is defined as follows. Given n terminals located at the same side of a fixed line l in the Euclidean plane ${\mathbb {R}}^2$ and a constant L, we are asked to find a Steiner tree T to interconnect these n terminals in ${\mathbb {R}}^2$ such that the Steiner points of the tree T, which has at least one Steiner point, are all located on the fixed line l and that the weight $w(T)=\sum _{e\in T}w(e) \le L$, the objective is to minimize the number s(T) of Steiner points of the tree T, where the weight $w(e)=0$ if the two endpoints of that edge $e\in T$ are located on the line l and otherwise the weight w(e) is the Euclidean distance between the two endpoints of that edge $e\in T$. In addition, when L is the minimum weight of all possible constrained 1-line-fixed Steiner trees as mentioned above, we refer to this version as the problem of minimum number of Steiner points of constrained 1-line-fixed minimum Steiner tree (abbreviated to the MNSP-C1LF-MST problem). We obtain two main results. (1) Using strategies of finding a minimum spanning tree with a degree constraint, we can design a 3-approximation algorithm in time $O(n^2\log n)$ to solve the MNSP-C1LF-ST problem. (2) Combining Delaunay triangulation properties and strategies of finding a minimum spanning tree with a degree constraint, we can provide a simple exact algorithm in time $O(n\log n \log \beta (n))$ to solve the MNSP-C1LF-MST problem, where $\beta (n)=\min \{i~|~\log ^{(i)} n \le 4-6/n\}$.