Capturing the Shape of a Point Set with a Line Segment

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.