On the Spanning Ratio of Theta-Graphs

We present improved upper bounds on the spanning ratio of a large family of θ-graphs. A θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m. We show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that θ-graphs with 4k + 3 and 4k + 5 cones have spanning ratio at most cos(θ/4)/(cos(θ/2)− sin(3θ/4)). This is a significant improvement on all families of θ-graphs for which exact bounds are not known. For example, the spanning ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. We also improve the upper bounds on the competitiveness of the θ-routing algorithm for these graphs to 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)) on θ-graphs with 4k + 4 cones and to 1 + 2 sin(θ/2) · cos(θ/4)/(cos(θ/2) − sin(3θ/4)) on θ-graphs with 4k + 3 and 4k + 5 cones. For example, the routing ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 4.0490.