New and Improved Spanning Ratios for Yao Graphs

For a set of points in the plane and a fixed integer k > 0, the Yao graph Yk 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 Y5, 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 Yk 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.