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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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jean-Lou De Carufel
    • 1
  • Carsten Grimm
    • 2
    • 3
  • 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

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