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.  ACM Subject Classification A network is a connected, undirected graph with positive edge weights. A geometric network is a network that is embedded in the Euclidean plane whose edges are rectifiable curves weighted with their length. We describe our results for straight-line edges, even though they extend to more general edges. We say that a point p lies on a geometric network G and write p ∈ G when there is an edge e of G such that p is a point along the embedding of e. A point p on an edge e of length l subdivides e into two sub-edges of lengths (1 − λ) · l and λ · l for some value λ ∈ [0, 1]. We represent the points on G in terms of their relative position (expressed by λ) along their containing edge, thereby avoiding ambiguity in case of crossings.
Trees have been studied in a recent follow-up work REF , which presents an algorithm to find an optimal shortcut for a tree of size n in O(n log n) time.
Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut
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