Papers › Shortcuts for the Circle
Shortcuts for the Circle
Sang Won Bae, Mark de Berg, Otfried Cheong, Joachim Gudmundsson, Christos Levcopoulos
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Let C be the unit circle in ℝ². 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 \emph{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(1/k^(2/3)) for any~k.
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