On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted $\paz$, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the $\paz$ graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the $\paz$ graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on $\paz$. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of $\paz$ is better than the proven theoretical bounds.