Given a point set $P$ in a metric space and a real number $t \geq 1$, an \emph{oriented $t$-spanner} is an oriented graph $\overrightarrow{G}=(P,\overrightarrow{E})$, where for every pair of distinct points $p$ and $q$ in $P$, the shortest oriented closed walk in $\overrightarrow{G}$ that contains $p$ and $q$ is at most a factor $t$ longer than the perimeter of the smallest triangle in $P$ containing $p$ and $q$. The \emph{oriented dilation} of a graph $\overrightarrow{G}$ is the minimum $t$ for which $\overrightarrow{G}$ is an oriented $t$-spanner. We present the first algorithm that computes, in Euclidean space, a sparse oriented spanner whose oriented dilation is bounded by a constant. More specifically, for any set of $n$ points in $\mathbb{R}^d$, where $d$ is a constant, we construct an oriented $(2+\varepsilon)$-spanner with $\mathcal{O}(n)$ edges in $\mathcal{O}(n \log n)$ time and $\mathcal{O}(n)$ space. Our construction uses the well-separated pair decomposition and an algorithm that computes a $(1+\varepsilon)$-approximation of the minimum-perimeter triangle in $P$ containing two given query points in $\mathcal{O}(\log n)$ time. While our algorithm is based on first computing a suitable undirected graph and then orienting it, we show that, in general, computing the orientation of an undirected graph that minimises its oriented dilation is NP-hard, even for point sets in the Euclidean plane. We further prove that even if the orientation is already given, computing the oriented dilation is APSP-hard for points in a general metric space. We complement this result with an algorithm that approximates the oriented dilation of a given graph in subcubic time for point sets in $\mathbb{R}^d$, where $d$ is a constant.

翻译：给定度量空间中的点集$P$和实数$t \geq 1$，\emph{定向$t$-生成子图}是指有向图$\overrightarrow{G}=(P,\overrightarrow{E})$，其中对于$P$中任意两个不同点$p$和$q$，$\overrightarrow{G}$中包含$p$和$q$的最短有向闭路径长度至多是$P$中包含$p$和$q$的最小三角形周长的$t$倍。图$\overrightarrow{G}$的\emph{定向膨胀度}是使$\overrightarrow{G}$成为定向$t$-生成子图的最小$t$值。本文提出了首个在欧氏空间中计算稀疏定向生成子图的算法，其定向膨胀度受常数界限制。具体而言，对于$\mathbb{R}^d$中任意$n$个点集（其中$d$为常数），我们可在$\mathcal{O}(n \log n)$时间和$\mathcal{O}(n)$空间内构造具有$\mathcal{O}(n)$条边的定向$(2+\varepsilon)$-生成子图。我们的构造方法采用良分离对分解技术，以及能在$\mathcal{O}(\log n)$时间内计算包含两个给定查询点的最小周长三角形$(1+\varepsilon)$近似解的算法。虽然我们的算法基于先计算合适的无向图再进行定向的思路，但我们证明：在一般情况下，即使对于欧氏平面中的点集，计算使定向膨胀度最小化的无向图定向问题是NP难的。我们进一步证明：即使在定向已给定的情况下，对于一般度量空间中的点集，计算定向膨胀度属于APSP难问题。作为补充，我们提出了在$\mathbb{R}^d$（$d$为常数）点集上于次立方时间内近似计算给定图定向膨胀度的算法。