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arXiv · 1612.02412

Shortcuts for the Circle

Sang Won Bae·Mark de Berg·Otfried Cheong·Joachim Gudmundsson·Christos Levcopoulos
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Abstract

Let $C$ be the unit circle in $\mathbb{R}^2$. 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 $k\geq 1$ \emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. We analyze for each $k$ with $1\leq k\leq 7$ 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 $2 + \Theta(1/k^{\frac{2}{3}})$ for any~$k$.

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Sang Won Bae, Mark de Berg, Otfried Cheong, Joachim Gudmundsson, Christos Levcopoulos. 2016-12-07. Shortcuts for the Circle. https://arxiv.org/abs/1612.02412

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