The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered $\theta$-graphs. A $\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\theta = 2 \pi/m$. An ordered $\theta$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \geq 1$, ordered $\theta$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(\theta/2) / (\cos(\theta/2) - \sin(\theta/2))$. We also show that for any integer $k \geq 2$, ordered $\theta$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(\theta/2))$. We provide lower bounds for ordered $\theta$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $\theta$-graphs with $4k + 2$ and $4k + 5$ cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered $\theta$-graphs have worse spanning ratios than unordered $\theta$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $\theta$-graphs with 4, 5, and 6 cones are not spanners.