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Online Routing in Directed Yao₄^∞ Graphs

Authors Prosenjit Bose , Jean-Lou De Carufel , John Stuart

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Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Geometric Spanners
  • Yao Graphs
  • Local Routing Algorithms

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Abstract

The x⃑{Yao₄^∞} and x⃑{Yao₄} graphs are two families of directed geometric graphs whose vertices are points in the plane, and each vertex has up to four outgoing edges. Consider a horizontal and a vertical line through each vertex v, defining four quadrants around v. Then v has an outgoing edge to the closest vertex in each of its four quadrants. When the distance is measured using the Euclidean norm, the resulting graph is the x⃑{Yao₄} graph, whereas with the L_∞-norm, we obtain the x⃑{Yao^{∞}₄} graph, which is a sub-graph of the well-known L_∞-Delaunay graph.
In this paper, we provide a local routing algorithm with routing ratio at most 85.22 for x⃑{Yao^{∞}₄} graphs. Prior to this work, no constant spanning or routing ratios for x⃑{Yao₄^∞} graphs were previously known. Now, x⃑{Yao₄^∞} graphs are the sparsest family of directed planar graphs supporting a competitive local routing strategy. Furthermore, we show that no local routing algorithm for x⃑{Yao₄^∞} graphs can have a routing ratio lower than 7+√2≈ 8.41. Moreover, we prove that the spanning ratio is at least 5+√2≈ 6.41 in the worst case. The techniques we develop in this paper also allow us to prove lower bounds of 7-√3+√{5-2√3}≈ 6.51 and 7+√2 for the spanning and routing ratios of x⃑{Yao₄}, respectively.

Cite As Get BibTex

Prosenjit Bose, Jean-Lou De Carufel, and John Stuart. Online Routing in Directed Yao₄^∞ Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.9

Author Details

Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
John Stuart
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada

Funding

This research is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

References

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