#### Document Type

Article

#### Publication Date

2014

#### Publication Title

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

#### 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 Y5, 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 Y5 from 10.9 to 2 + √3 ≈ 3.74, which falls slightly below the lower bound of 3.79 established for the spanning ratio of ⊝5 (⊝-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 ⊝-graph with the same number of cones. We also give a lower bound of 2.87 on the spanning ratio of Y5. Finally, we revisit the Y6 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 ≤ 6) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of Y6 from 17.6 to 5.8, getting closer to the spanning ratio of 2 established for ⊝6.

#### DOI

dx.doi.org/10.1145/2582112.2582143

#### Recommended Citation

Barba, Luis; Bose, Prosenjit; Damian, Mirela; Fagerberg, Rolf; Keng, Wah Loon; O'Rourke, Joseph; van Renssen, André; Taslakian, Perouz; Verdonschot, Sander; and Xia, Ge, "New and Improved Spanning Ratios for Yao Graphs" (2014). *Computer Science: Faculty Publications*. 23.

https://scholarworks.smith.edu/csc_facpubs/23

## Comments

Peer reviewed post-print. ISBN: 978-1-4503-2594-3