Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line segment. Specifically, given a radius $r$, we say a line segment is representative of a point set $P$ if it is within distance $r$ of each point $p \in P$. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius $r$ using the shortest line segment. We describe an algorithm to find the shortest representative segment in $O(n \log h + h \log^3 h)$ time. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in $P$ move.

翻译：检测点集中位置相关群组是众多应用领域中的重要任务。除了简单检测这些群组外，其形状也承载着意义。本文采用简单几何对象——线段来表示群组形状。具体而言，给定半径 $r$，若一条线段与点集 $P$ 中每个点 $p \in P$ 的距离均不超过 $r$，则称该线段可表示点集 $P$。我们的目标是寻找最短的此类线段。该问题等价于用最短线段刺穿一组半径为 $r$ 的圆。我们提出一种时间复杂度为 $O(n \log h + h \log^3 h)$ 的算法来寻找最短代表线段。此外，我们展示了在点集 $P$ 移动时如何保持最短代表线段的稳定近似。