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.  \approx 17.03.}
We present a routing algorithm for the directed \Theta_4-graph, here denoted as the \overrightarrow{\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 \sqrt{290 \approx 17.03.}
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Theta-4-graph
directed \Theta_4-graph, here denoted as the \overrightarrow{\Theta_4
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Theta-4-graph
\Theta_4-graph
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We also show that without this additional bit of information, the routing ratio increases to \sqrt{290