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

• Published in: Computational geometry : theory and applications Vol. 105-106; p. 101881
• Main Authors: Akitaya, Hugo A., Biniaz, Ahmad, Bose, Prosenjit
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-08-2022
• Subjects: Online routing; Routing ratio; Spanning ratio; Theta graphs; Theta graphs; Online routing; Spanning ratio; Routing ratio
• ISSN: 0925-7721
• DOI: 10.1016/j.comgeo.2022.101881

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.