WADS 2017: Algorithms and Data Structures pp 301-312

# Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut

• Jean-Lou De Carufel
• Carsten Grimm
• Stefan Schirra
• Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## 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.

## Keywords

Network augmentation Continuous diameter minimization

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© Springer International Publishing AG 2017

## Authors and Affiliations

• Jean-Lou De Carufel
• 1
• Carsten Grimm
• 2
• 3
Email author
• Stefan Schirra
• 3
• Michiel Smid
• 2
1. 1.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada
2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
3. 3.Institut für Simulation und GraphikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany