Minimizing the Continuous Diameter when Augmenting a Geometric Tree with a Shortcut

TL;DR

This paper presents an efficient method to add a single shortcut to a geometric tree in the Euclidean plane to minimize its continuous diameter, with a characterization of when such a shortcut is effective.

Contribution

We provide a necessary and sufficient condition for the existence of a beneficial shortcut and develop an $O(n \,\log\, n)$ algorithm to find an optimal one for geometric trees.

Findings

A single shortcut reduces the continuous diameter iff the intersection of all diametral paths is neither a line segment nor a point.

The optimal shortcut can be found in $O(n \log n)$ time for trees with straight-line edges.

Results extend to trees with rectifiable curved edges.

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, 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 $O(n \log n)$ 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.