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Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

  • Nicolas Bonichon
  • Prosenjit Bose
  • Paz Carmi
  • Irina Kostitsyna
  • Anna LubiwEmail author
  • Sander Verdonschot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

A geometric graph is angle-monotone if every pair of vertices has a path between them that—after some rotation—is x- and y-monotone. Angle-monotone graphs are \(\sqrt{2}\)-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone—specifically, we prove that the half-\(\theta _6\)-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex s to any vertex t whose length is within \(1 + \sqrt{2}\) times the Euclidean distance from s to t. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.

Notes

Acknowledgements

This work was begun at the CMO-BIRS Workshop on Searching and Routing in Discrete and Continuous Domains, October 11–16, 2015. We thank the other participants of the workshop for many good ideas and stimulating discussions. We thank an anonymous referee for helpful comments.

Funding acknowledgements: A.L. thanks NSERC (Natural Sciences and Engineering Council of Canada). S.V. thanks NSERC and the Ontario Ministry of Research and Innovation. N.B. thanks French National Research Agency (ANR) in the frame of the “Investments for the future” Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02). I.K. was supported in part by the NWO under project no. 612.001.106, and by F.R.S.-FNRS.

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Nicolas Bonichon
    • 1
  • Prosenjit Bose
    • 2
  • Paz Carmi
    • 3
  • Irina Kostitsyna
    • 4
  • Anna Lubiw
    • 5
    Email author
  • Sander Verdonschot
    • 6
  1. 1.LaBRIUniversity of BordeauxBordeauxFrance
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Department of Computer ScienceBen-Gurion University of the NegevBeershebaIsrael
  4. 4.Université Libre de BruxellesBrusselsBelgium
  5. 5.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  6. 6.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada

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