Abstract
The family of Θk-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 Θ6-graph. We show that the directed Θ6-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 8 and show that no algorithm can have a routing ratio better than 6−ε.
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