On the Spanning and Routing Ratio of $\Theta_4$
Darryl Hill
Carleton University

We present a routing algorithm for the $\Theta_4$-graph that computes a path between any two vertices $s$ and $t$ having length at most 17 times the Euclidean distance between $s$ and $t$. To compute this path, at each step, the algorithm only uses knowledge of the location of the current vertex, its (at most four) outgoing edges, the destination vertex, and one additional bit of information in order to determine the next edge to follow. This provides the first known online, local, competitive routing algorithm with constant routing ratio for the $\Theta_4$-graph, as well as improving the best known upper bound on the spanning ratio of these graphs from 237 to 17. We also show that without this additional bit of information, the routing ratio increases to the square root of 290, which is just under 17.03.