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：给定欧几里得平面上的点集$P$和参数$t$，我们将\emph{定向$t$-生成器}定义为完全双向图的一个定向子图，使得对于任意一对点，$G$中经过这两点的最短有向环长度至多是完全双向图中最短有向环长度的$t$倍。我们研究计算具有小定向扩张的稀疏图的问题。由于我们能够证明在平面上给定边数条件下最小化定向扩张是NP难的，因此我们首先考虑一维点集。虽然在此情形下获得$1$-生成器是直接的，但即使对于五个点，这样的生成器也不能存在一种将最左端点和最右端点置于外表面的平面嵌入。这导致我们限制考虑在一维点集上具有单页书式嵌入的定向图。针对这种情形，我们提出一种动态规划算法，能够在$O(n^8)$时间内计算出$n$个点的最小定向扩张图，以及一种贪心算法，能够在$O(n\log n)$时间内计算出$5$-生成器。扩展这些结果最终得到二维点集上的结论：我们证明对于凸点集，贪心三角剖分能够生成一个定向$O(1)$-生成器。