Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut

Abstract

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, 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 $T+pq$. The continuous diameter is the largest distance between any two points along edges. 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 point. We determine an optimal shortcut for a geometric tree with $n$ straight-line edges in $O(n \log n)$ time.

This work was partially supported by NSERC and FQRNT.