窄条上的直线Steiner树
Rectilinear Steiner Trees in Narrow Strips
摘要：
用于$\mathbb{R}^2$中的点的集合$P$的直线斯坦纳树是使用水平和垂直线段连接$P$中的点的树。Minimal Rectilinear Steiner Tree的目标是找到一棵总长度最小的Steiner树。研究了带$(-\infty,+\infty)\times [0,δ]$内点集$P$的最小矩形Steiner树的复杂性如何依赖于带的宽度$δ$ .我们得到两个主要结果。1)我们提出了一个算法的运行时间$n^{O(\sqrtδ)}$的稀疏点集，即点集，其中每个$1\timesδ$矩形内的带包含$O(1)$点。2)对于随机点集，其中的点是随机选择的高度$δ$和预期宽度$n$的矩形内，我们提出了一种算法，是固定参数易于处理的$δ$和线性$n$。它的预期运行时间为$2^{O(δ\sqrtδ)} n$。
英文摘要：
A rectilinear Steiner tree for a set $P$ of points in $\mathbb{R}^2$ is a tree that connects the points in $P$ using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of Minimal Rectilinear Steiner Tree for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,δ]$ depends on the strip width $δ$. We obtain two main results. 1) We present an algorithm with running time $n^{O(\sqrtδ)}$ for sparse point sets, that is, point sets where each $1\timesδ$ rectangle inside the strip contains $O(1)$ points. 2) For random point sets, where the points are chosen randomly inside a rectangle of height $δ$ and expected width $n$, we present an algorithm that is fixed-parameter tractable with respect to $δ$ and linear in $n$. It has an expected running time of $2^{O(δ\sqrtδ)} n$.
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