Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut

We augment a tree T with a shortcut pq to minimize the largest distance between any two points along the resulting augmented tree T + pq. We study this problem in a continuous and geometric setting where T is a geometric tree in the Euclidean plane, where a shortcut is a line segment connecting any two points along the edges of T , and we consider all points on T + pq (i.e., vertices and points along edges) when determining the largest distance along T + pq. We refer to the largest distance between any two points along edges as the continuous diameter to distinguish it from the discrete diameter, i.e., the largest distance between any two vertices. We establish that a single shortcut is sufficient to reduce the continuous diameter of a geometric tree T if and only if the intersection of all diametral paths of T is neither a line segment nor a single point. We determine an optimal shortcut for a geometric tree with n straight-line edges in O(n logn) time. Apart from the running time, our results extend to geometric trees whose edges are rectifiable curves. The algorithm for trees generalizes our algorithm for paths. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory, I.3.5 Computational Geometry and Object Modeling