A Linear-Time Algorithm for Discrete Radius Optimally Augmenting Paths in a Metric Space

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of $P$. Previously, the "continuous" version of the problem where a center may be a point in the interior of an edge of the graph was studied and a linear-time algorithm was known. Our "discrete" version of the problem has not been studied before. We present a linear-time algorithm for the problem.