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

Abstract

Let [Formula: see text] be a path graph of [Formula: see text] vertices embedded in a metric space. We consider the problem of adding a new edge to [Formula: see text] so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of [Formula: see text]. 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.

Algorithms for Diameters of Unicycle Graphs and Diameter-Optimally Augmenting Trees

“…The radius considered in [17] is defined with respect to all points of T , not just the vertices. Wang and Zhao [24] studied the same problem with radius defined with respect to only the vertices, and they gave a linear time algorithm.…”