Expected complexity of routing in $\Theta_6$ and half-$\Theta_6$ graphs

Abstract

We study online routing algorithms on the $\Theta_6$-graph and the half-$\Theta_6$-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex $s$ and a target vertex $t$ in the $\Theta_6$-graph (resp. half-$\Theta_6$-graph), there exists a deterministic online routing algorithm that finds a path from $s$ to $t$ whose length is at most $2\|st\|$ (resp. $2.89\|st\|$) which is optimal in the worst case [Bose et al, SIAM J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $\Theta_6$-graph and half-$\Theta_6$-graph defined on a Poisson point process.

For the $\Theta_6$-graph,  our online routing algorithm has an expected routing factor of $1.161$ when $s$ and $t$ are random. The routing factor is the length of the route between $s$ and $t$ produced by our algorithm divided by the Euclidean distance between $s$ and $t$. Moreover, our routing algorithm has a maximum expected routing factor of $1.22$, where the maximum is for fixed $s$ and $t$ and all other points are random. This is much better than the worst-case routing ratio of $2$. The routing ratio is the maximum routing factor among all pairs of points. For the half-$\Theta_6$-graph, our memoryless online routing algorithm has an expected routing factor of $1.43$ and a maximum expected routing factor of $1.58$. Our online routing algorithm that uses a constant amount of additional  memory has an expected routing factor of $1.34$ and a maximum expected routing factor of $1.40$. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing factor that is much better than their worst-case routing ratio of $2.89$.