Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in $G$ through those points is at most a factor $t$ longer than the shortest oriented cycle in the complete bi-directed graph. 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 $O(n^8)$ 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 an oriented $O(1)$-spanner.

翻译：给定欧几里得平面中的点集 $P$ 和参数 $t$，我们将 \emph{有向 $t$-稀疏图}定义为完全双向图的有向子图，使得对于任意一对点，在 $G$ 中经过这两点的最短环的长度最多比完全双向图中的最短有向环长 $t$ 倍。我们研究计算具有小有向扩张度的稀疏图的问题。由于可以证明在平面中给定边数最小化有向扩张度是 NP-难的，我们首先考虑一维点集。虽然在此设置中获取 $1$-稀疏图是直接的，但对于五个点而言，这样的稀疏图已无法将最左侧和最右侧的点嵌入到外平面的平面图上。这导致我们限制为在一维点集上具有单页书式嵌入的有向图。针对这种情况，我们提出了一种动态规划算法，可在 $O(n^8)$ 时间内计算 $n$ 个点的最小有向扩张度图，以及一种贪心算法，可在 $O(n\log n)$ 时间内计算 $5$-稀疏图。最后，将这些结果扩展到二维点集：我们证明对于凸点集，贪心三角剖分产生一个有向 $O(1)$-稀疏图。