The key insights and highlights of the content are:
Detecting location-correlated groups or clusters in point sets is an important task in various applications. The shape of these groups can carry meaningful information, and a simple geometric object like a line segment can be used to represent the group's shape.
The authors propose a model where the goal is to find the shortest line segment q1q2 such that all points in the point set P are within distance r of the segment. This model has several advantages, including naturally handling the case where no line segment represents the points or a single point already represents the points.
The authors present an algorithm to find the shortest representative line segment in O(n log h + h log^3 h) time, where n is the number of points and h is the size of the convex hull of P. The algorithm uses a rotating calipers approach, maintaining various geometric structures and handling different types of events during the rotation.
For the kinetic version of the problem, where the points move, the authors show how to maintain a stable approximation of the shortest representative segment, where the endpoints move with bounded speed and the segment does not flicker rapidly between "on" and "off".
The authors build upon existing results in computational geometry, such as the combinatorial result on the number of conjugate pairs in a convex polygon, to achieve their efficiency bounds.
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arxiv.org
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