On the Spanning and Routing Ratio of Directed Theta-Four

TL;DR

This paper introduces the first online, local routing algorithm for directed Theta-4 graphs with a constant routing ratio, significantly improving the known spanning ratio bounds from 237 to 17.

Contribution

It presents a novel routing algorithm for directed Theta-4 graphs that achieves a constant routing ratio and improves the upper bound on the spanning ratio.

Findings

Routing ratio is at most 17 with the new algorithm.

Without extra information, routing ratio slightly exceeds 17, approximately 17.03.

First online, local routing algorithm with constant ratio for directed Theta-4 graphs.

Abstract

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 $\sqrt{290} \approx 17.03$.