Advertisement

Upper and Lower Bounds for Online Routing on Delaunay Triangulations

  • Nicolas Bonichon
  • Prosenjit Bose
  • Jean-Lou De Carufel
  • Ljubomir Perković
  • André van Renssen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)

Abstract

Consider a weighted graph G whose vertices are points in the plane and edges are line segments between pairs of points whose weight is the Euclidean distance between its endpoints. A routing algorithm on G sends a message from any vertex s to any vertex t in G. The algorithm has a competitive ratio of c if the length of the path taken by the message is at most c times the length of the shortest path from s to t in G. It has a routing ratio of c if the length of the path is at most c times the Euclidean distance from s to t. The algorithm is online if it makes forwarding decisions based on (1) the k-neighborhood in G of the message’s current position (for constant k > 0) and (2) limited information stored in the message header.

We present an online routing algorithm on the Delaunay triangulation with routing ratio less than 5.90, improving the best known routing ratio of 15.48. Our algorithm makes forwarding decisions based on the 1-neighborhood of the current position of the message and the positions of the message source and destination only.

We present a lower bound of 5.7282 on the routing ratio of our algorithm, so the 5.90 upper bound is close to the best possible. We also show that the routing (resp., competitive) ratio of any deterministic k-local algorithm is at least 1.70 (resp., 1.23) for the Delaunay triangulation and 2.70 (resp. 1.23) for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there always exists a path between s and t whose length is at most 2.61|[st]|, it is not always possible to route a message along a path of length less than 2.70|[st]| using only local information.

Keywords

Delaunay triangulation online routing routing ratio competitive ratio 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonichon, N., Gavoille, C., Hanusse, N., Perković, L.: The stretch factor of l 1- and l  ∞ -delaunay triangulations. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 205–216. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Bose, P., Morin, P.: Online routing in triangulations. SIAM J. Comp. 33(4), 937–951 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bose, P., Smid, M.: On plane geometric spanners: A survey and open problems. Comput. Geom. 46(7), 818–830 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bose, P., De Carufel, J.L., Durocher, S., Taslakian, P.: Competitive online routing on Delaunay triangulations. In: SWAT, pp. 98–109 (2014)Google Scholar
  5. 5.
    Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Competitive routing in the half-θ 6-graph. In: SODA, pp. 1319–1328 (2012)Google Scholar
  6. 6.
    Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Competitive routing on a bounded-degree plane spanner. In: CCCG, pp. 299–304 (2012)Google Scholar
  7. 7.
    Braunl, T.: Embedded Robotics: Mobile Robot Design and Applications with Embedded Systems. Springer (2006)Google Scholar
  8. 8.
    Broutin, N., Devillers, O., Hemsley, R.: Efficiently navigating a random Delaunay triangulation. In: AofA, pp. 49–60 (2014)Google Scholar
  9. 9.
    Chew, L.P.: There is a planar graph almost as good as the complete graph. In: SoCG, pp. 169–177 (1986)Google Scholar
  10. 10.
    Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comp. System Sci. 39(2), 205–219 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. Discrete & Comput. Geom. 5(4), 399–407 (1990). doi:10.1007/BF02187801Google Scholar
  13. 13.
    Hofmann-Wellenhof, B., Legat, K., Wieser, M.: Navigation: Principles of Positioning and Guidance. Springer (2003)Google Scholar
  14. 14.
    Murgante, B., Borruso, G., Lapucci, A.: Geocomputation and Urban Planning. In: Murgante, B., Borruso, G., Lapucci, A. (eds.) Geocomputation and Urban Planning. SCI, vol. 176, pp. 1–17. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Narasimhan, G., Smid, M.H.M.: Geometric spanner networks. Cambridge University Press (2007)Google Scholar
  16. 16.
    Stojmenovic, I.: Handbook of Wireless Networks and Mobile Computing. Wiley-Interscience (2002)Google Scholar
  17. 17.
    Worboys, M.F., Duckham, M.: GIS: A Computing Perspective, 2nd edn. CRC Press (2004)Google Scholar
  18. 18.
    Xia, G.: The stretch factor of the Delaunay triangulation is less than 1. 998. SIAM J. Comput. 42(4), 1620–1659 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Xia, G., Zhang, L.: Toward the tight bound of the stretch factor of Delaunay triangulations. In: CCCG, pp. 175–180 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Nicolas Bonichon
    • 1
    • 2
  • Prosenjit Bose
    • 3
  • Jean-Lou De Carufel
    • 3
  • Ljubomir Perković
    • 4
  • André van Renssen
    • 5
    • 6
  1. 1.Univ. Bordeaux, LaBRI, UMR 5800TalenceFrance
  2. 2.CNRS, LaBRI, UMR 5800TalenceFrance
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada
  4. 4.School of ComputingDePaul UniversityChicagoUSA
  5. 5.National Institute of InformaticsTokyoJapan
  6. 6.JST, ERATO, Kawarabayashi Large Graph ProjectTokyoJapan

Personalised recommendations