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=2 π/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.