Shortcuts for the Circle

TL;DR

This paper investigates how adding a limited number of shortcuts on a circle can minimize the graph's diameter, revealing non-monotonic improvements and asymptotic behavior as shortcuts increase.

Contribution

It characterizes optimal shortcut placements for up to seven shortcuts and proves the asymptotic diameter reduction as shortcuts grow.

Findings

Optimal shortcuts for k=1 to 7 are identified.

Adding more than six shortcuts does not always reduce the diameter.

Diameter approaches 2 at a rate of Θ(1/k^{2/3}) as k increases.

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~$k$.