# New and Improved Spanning Ratios for Yao Graphs

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

January 2014

### Abstract

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 $ Θ =2π/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 $k ≥ 5$, the spanning ratio of $Y_k$ is at most $1/(1-2sin(3Θ/8))$, which gives the first constant upper bound for $Y_5$, and is an improvement over the previous bound of $1/(1-2sin(Θ/2))$ for odd $k ≥ 7$. We further reduce the upper bound on the spanning ratio for $Y_5$ from $10.9$ to $2+ sqrt(3) ≈ 3.74$, which falls slightly below the lower bound of $3.79$ established for the spanning ratio of $Θ_5$ ($Θ$-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 $Θ$-graph with the same number of cones. We also give a lower bound of $2.87$ on the spanning ratio of $Y_5$. In addition, we revisit the $Y_6$ graph, which plays a particularly important role as the transition between the graphs ($k > 6$) for which simple inductive proofs are known, and the graphs ($k łeq 6$) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of $Y_6$ from $17.6$ to $5.8$, getting closer to the spanning ratio of 2 established for $Θ_6$. Finally, we present the first lower bounds on the spanning ratio of Yao graphs with more than six cones, and a construction that shows that the Yao-Yao graph (a bounded-degree variant of the Yao graph) with five cones is not a spanner.

Publication

*Journal of Computational Geometry*