Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} $G$ as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest closed walk in $G$ through those points is at most a factor $t$ longer than the shortest cycle in the complete graph on $P$. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a $1$-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $O(n^7)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in a plane oriented $t$-spanner with $t=19 \cdot t_g$, where $t_g$ is a upper bound on the dilation of the greedy triangulation.

翻译：给定欧几里得平面上的点集 $P$ 及参数 $t$，我们将\emph{定向 $t$-生成树} $G$ 定义为完全双向图的一个有向子图，使得对于任意点对，$G$ 中经过这些点的最短闭合路径长度至多是 $P$ 上完全图的最短环长度的 $t$ 倍。我们研究了计算具有较小定向扩张系数的稀疏图的问题。由于我们能够证明在平面上对给定边数最小化定向扩张系数是 NP 难问题，我们首先考虑一维点集。虽然在此设定下获得 $1$-生成树较为直接，但即使对于五个点，此类生成树也不存在最左与最右点同时位于外表面的平面嵌入。这导致我们将研究限制在具有一维点集上单页书嵌入的定向图。针对这种情况，我们提出了一种动态规划算法以计算最小定向扩张系数的图，该算法在 $n$ 个点上运行时间为 $O(n^7)$，同时给出了一种在 $O(n\log n)$ 时间内计算 $5$-生成树的贪心算法。扩展这些结果最终为我们提供了二维点集的结论：我们证明对于凸点集，贪心三角剖分可产生平面定向 $t$-生成树，其中 $t=19 \cdot t_g$，而 $t_g$ 是贪心三角剖分扩张系数的上界。