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DOI: 10.4230/LIPIcs.ISAAC.2024.15

On the Spanning and Routing Ratios of the Yao-Four Graph

Authors: Prosenjit Bose, Darryl Hill, Michiel Smid, and Tyler Tuttle

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

The Yao graph is a geometric spanner that was independently introduced by Yao [SIAM J. Comput., 1982] and Flinchbaugh and Jones [SIAM J. Algebr. Discret. Appl., 1981]. We prove that for any two vertices of the undirected version of the Yao graph with four cones, there is a path between them with length at most 13 + 5/√2 ≈ 16.54 times the Euclidean distance between the vertices, improving the previous best bound of approximately 54.62. We also present an online routing algorithm for the directed Yao graph with four cones that constructs a path between any two vertices with length at most 17 + 9/√2 ≈ 23.36 times the Euclidean distance between the vertices. This is the first routing algorithm for a directed Yao graph with fewer than six cones. The algorithm uses knowledge of the coordinates of the current vertex, the (up to) four neighbours of the current vertex, and the destination vertex to make a routing decision. It also uses one additional bit of memory. We show how to dispense with this single bit at the cost of increasing the length of the path to √{331 + 154√2} ≈ 23.43 times the Euclidean distance between the vertices.

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Prosenjit Bose, Darryl Hill, Michiel Smid, and Tyler Tuttle. On the Spanning and Routing Ratios of the Yao-Four Graph. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bose_et_al:LIPIcs.ISAAC.2024.15,
  author =	{Bose, Prosenjit and Hill, Darryl and Smid, Michiel and Tuttle, Tyler},
  title =	{{On the Spanning and Routing Ratios of the Yao-Four Graph}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.15},
  URN =		{urn:nbn:de:0030-drops-221422},
  doi =		{10.4230/LIPIcs.ISAAC.2024.15},
  annote =	{Keywords: Yao graph, online routing, geometric spanners}
}

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On the Spanning and Routing Ratios of the Yao-Four Graph

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