Shortcuts for the circle

Sang Won Bae, Mark de Berg, Otfried Cheong (Corresponding author), Joachim Gudmundsson, Christos Levcopoulos

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Let C be the unit circle in R 2 . We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k⩾1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1⩽k⩽7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2+Θ([formula presented]) for any k.

Original languageEnglish
Pages (from-to)37-54
Number of pages18
JournalComputational Geometry
Volume79
DOIs
Publication statusPublished - 1 Feb 2019

Keywords

  • Geometric network
  • Graph augmentation
  • Graph diameter

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