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On the Spanning and Routing Ratio of the Directed Theta-Four Graph

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Abstract

We present a routing algorithm for the directed \(\Theta _4\)-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 show that 17 is the lowest routing ratio this algorithm will achieve on this graph. We also show that we can easily remove this additional bit of information without additional complexity and the routing ratio increases to \(\sqrt{290}\) which is approximately 17.03.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Dr., Ottawa, ON, K1S 5B6, Canada

    Prosenjit Bose, Darryl Hill & Michiel Smid

  2. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Ave., Ottawa, ON, K1N 6N5, Canada

    Jean-Lou De Carufel

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  1. Prosenjit Bose
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  2. Jean-Lou De Carufel
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  4. Michiel Smid
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Correspondence to Darryl Hill.

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Bose, P., De Carufel, JL., Hill, D. et al. On the Spanning and Routing Ratio of the Directed Theta-Four Graph. Discrete Comput Geom 71, 872–892 (2024). https://doi.org/10.1007/s00454-023-00597-8

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