We study the problem of ordered stabbing of $n$ balls (of arbitrary and possibly different radii, no ball contained in another) in $\mathbb{R}^d$, $d \geq 3$, with either a directed line segment or a (directed) polygonal curve. Here, the line segment, respectively polygonal curve, shall visit (intersect) the given sequence of balls in the order of the sequence. We present a deterministic algorithm that decides whether there exists a line segment stabbing the given sequence of balls in order, in time $O(n^{4d-2} \log n)$. Due to the descriptional complexity of the region containing these line segments, we can not extend this algorithm to actually compute one. We circumvent this hurdle by devising a randomized algorithm for a relaxed variant of the ordered line segment stabbing problem, which is built upon the central insights from the aforementioned decision algorithm. We further show that this algorithm can be plugged into an algorithmic scheme by Guibas et al., yielding an algorithm for a relaxed variant of the minimum-link ordered stabbing path problem that achieves approximation factor 2 with respect to the number of links. We conclude with experimental evaluations of the latter two algorithms, showing practical applicability.

翻译：我们研究了在 $\mathbb{R}^d$（$d \geq 3$）中有序刺穿 $n$ 个球体（半径任意且可能不同，且无球包含于另一球内）的问题，刺穿工具为有向线段或（有向）多边形曲线。此处，线段或多边形曲线需按给定顺序依次访问（相交）这批球体。我们提出一种确定性算法，可在 $O(n^{4d-2} \log n)$ 时间内判断是否存在一条线段能按顺序刺穿给定球体序列。由于包含这些线段的区域存在描述复杂度，我们无法将该算法扩展至实际计算出一条线段。为克服这一障碍，我们针对有序线段刺穿问题的一个松弛变体设计了一种随机算法，该算法基于前述决策算法的核心见解构建。进一步地，我们证明该算法可嵌入 Guibas 等人提出的算法框架，从而针对最小链接有序刺穿路径问题的一个松弛变体，得到一种在链接数上实现近似因子为 2 的算法。最后，我们对后两种算法进行了实验评估，展示了其实用适用性。