Let $G=(V,E)$ be an edge weighted geometric graph (not necessarily planar) such that every edge is horizontal or vertical. The weight of an edge $uv\in E$ is the $L_1$-distance between its endpoints. Let $W_G(u,v)$ denotes the length of a shortest path between a pair of vertices u and v in G. The graph G is said to be a Manhattan network for a given point set P in the plane if $P\subseteq V$ and $\forall p,q\in P$, $W_G(p,q)={\Vert pq\Vert }_1$. In addition to P, the graph G may also include a set T of Steiner points in its vertex set V. In the Manhattan network problem, the objective is to construct a Manhattan network of small size (the number of Steiner points) for a set of n points. This problem was first considered by Gudmundsson et al. [EuroCG 2007]. They give a construction of a Manhattan network of size $\mathrm{\Theta }(n\log n)$ for general point sets in the plane. We say a Manhattan network is planar if it has a planar embedding. In this paper, we construct a planar Manhattan network for convex point sets in linear time using $\mathcal{O}(n)$ Steiner points. We also show that, even for convex point sets, the construction in Gudmundsson et al. [EuroCG 2007] needs $\mathrm{\Omega }(n\log n)$ Steiner points, and the network may not be planar.

让 $G=(伏,乙)$是一个边加权几何图（不一定是平面），这样每条边都是水平或垂直的。边的权重$你v\in 乙$ 是个 ${升}_1$- 端点之间的距离。让${宽}_G(你,v)$表示最短路径的一对顶点之间的长度ù和v在ģ。对于平面中的给定点集P，图G被称为曼哈顿网络，如果$磷\subseteq 伏$ 和 $\forall 磷,q\in 磷$, ${宽}_G(磷,q)={\Vert 磷q\Vert }_1$. 除了P，图表ģ还可以包括一组Ť的斯坦纳点在其顶点集V。在曼哈顿网络问题中，目标是为一组n个点构建一个小规模（Steiner 点的数量）的曼哈顿网络。Gudmundsson 等人首先考虑了这个问题。[EuroCG 2007]。他们给出了一个曼哈顿规模的网络的构建$\mathrm{\Theta }(n\mathrm{日志}n)$用于平面中的一般点集。我们说曼哈顿网络是平面的，如果它有一个平面嵌入。在本文中，我们使用线性时间为凸点集构建平面曼哈顿网络$\mathcal{哦}(n)$施泰纳点。我们还表明，即使对于凸点集，Gudmundsson 等人的构造也是如此。[EuroCG 2007] 需要$\mathrm{\Omega }(n\mathrm{日志}n)$ Steiner 点，并且网络可能不是平面的。