Can I read New and Improved Spanning Ratios for Yao Graphs on EtoBox?

New and Improved Spanning Ratios for Yao Graphs by Barba, Luis; Bose, Prosenjit; Damian, Mirela; Fagerberg, Rolf; Keng, Wah Loon; O'Rourke, Joseph; van Renssen, André; Taslakian, Perouz; Verdonschot, Sander; Xia, Ge is a scholarly article available to read on EtoBox.

What is New and Improved Spanning Ratios for Yao Graphs about?

For a set of points in the plane and a fixed integer $k > 0$, the Yao graph $Y_k$ partitions the space around each point into $k$ equiangular cones of angle $\theta=2\pi/k$, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of $Y_5$, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd $k \geq 5$, the spanning ratio of $Y_k$ is at most $1/(1-2\sin(3\theta/8))$, which gives the first constant upper bound for $Y_5$, and is an improvement over the previous bound of $1/(1-2\sin(\theta/2))$ for odd $k \geq 7$. We further reduce the upper bound on the spanning ratio for $Y_5$ from $10.9$ to $2+\sqrt{3} \approx 3.74$, which falls slightly below the lower bound of $3.79$ established for the spanning ratio of $\Theta_5$ ($\Theta$-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and $\Theta$-graph with the same number of cones. We also give a lower bound of $2.87$ on the spanning ratio of $Y_5$. Finally, we revisit the $Y_6$ graph, which plays a particularly important role as the transitio

Author

Barba, Luis; Bose, Prosenjit; Damian, Mirela; Fagerberg, Rolf; Keng, Wah Loon; O'Rourke, Joseph; van Renssen, André; Taslakian, Perouz; Verdonschot, Sander; Xia, Ge

Published

2013

Language

EN