The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered $θ$-graphs. A $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$. An ordered $θ$-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 $θ$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(θ/2) / (\cos(θ/2) - \sin(θ/2))$. We also show that for any integer $k \geq 2$, ordered $θ$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(θ/2))$. We provide lower bounds for ordered $θ$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $θ$-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 $θ$-graphs have worse spanning ratios than unordered $θ$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $θ$-graphs with 4, 5, and 6 cones are not spanners.

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