On the spanning and routing ratios of the directed Θ6-graph

The family of Θ k -graphs is an important class of sparse geometric spanners with a small spanning ratio. Although they are a well-studied class of geometric graphs, no bound is known on the spanning and routing ratios of the directed Θ 6 -graph. We show that the directed Θ 6 -graph of a point set P , denoted Θ → 6 ( P ) , is a 7-spanner and there exist point sets where the spanning ratio is at least 4 - ε , for any ε > 0 . It is known that the standard greedy Θ -routing algorithm may have an unbounded routing ratio on Θ → 6 ( P ) . We design a simple, online, local, memoryless routing algorithm on Θ → 6 ( P ) whose routing ratio is at most 14 and show that no algorithm can have a routing ratio better than 6 - ε .