Towards Tight Bounds on ThetaGraphs
Abstract
We present improved upper and lower bounds on the spanning ratio of $\theta$graphs with at least six cones. Given a set of points in the plane, a $\theta$graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\theta=2\pi/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $\theta$graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(\theta/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $\theta$graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(\theta/2)/(\cos(\theta/2)\sin(\theta/2))$. We also show that $\theta$graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(\theta/4)/(\cos(\theta/2)\sin(3\theta/4))$. This is a significant improvement on all families of $\theta$graphs for which exact bounds are not known. For example, the spanning ratio of the $\theta$graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $\theta$routing algorithm. In particular, we show that the $\theta$routing algorithm is $(1+2\sin(\theta/2)/(\cos(\theta/2)\sin(\theta/2)))$competitive on $\theta$graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $\theta$graphs. In particular, we show that $\theta$graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(\theta/2)+2\tan^2(\theta/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $\theta$graphs with $4k+4$ cones is greater than that of $\theta$graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.6233
 Bibcode:
 2014arXiv1404.6233B
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 arXiv admin note: text overlap with arXiv:1401.2127