Local Routing in Spanners Based on WSPDs

  • Prosenjit Bose
  • Jean-Lou De Carufel
  • Vida Dujmović
  • Frédérik Paradis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

The well-separated pair decomposition (WSPD) of the complete Euclidean graph defined on points in \(\mathbb {R}^2\) (Callahan and Kosaraju [JACM, 42 (1): 67-90, 1995]) is a technique for partitioning the edges of the complete graph based on length into a linear number of sets. Among the many different applications of WSPDs, Callahan and Kosaraju proved that the sparse subgraph that results by selecting an arbitrary edge from each set (called WSPD-spanner) is a \(1+8/(s-4)\)-spanner, where \(s>4\) is the separation ratio used for partitioning the edges.

Although competitive local-routing strategies exist for various spanners such as Yao-graphs, \(\varTheta \)-graphs, and variants of Delaunay graphs, few local-routing strategies are known for any WSPD-spanner. Our main contribution is a local-routing algorithm with a near-optimal competitive routing ratio of \(1+O(1/s)\) on a WSPD-spanner. Specifically, we present a 2-local and a 1-local routing algorithm on a WSPD-spanner with competitive routing ratios of \(1+6/(s-2)+4/s\) and \(1+6/(s-2)+6/s+4/(s^2-2s)+8/{s^2}\), respectively.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonichon, N., Bose, P., De Carufel, J.-L., Perković, L., van Renssen, A.: Upper and lower bounds for online routing on Delaunay triangulations. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 203–214. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48350-3_18 CrossRefGoogle Scholar
  2. 2.
    Bose, P., Devroye, L., Löffler, M., Snoeyink, J., Verma, V.: Almost all Delaunay triangulations have stretch factor greater than pi/2. Comput. Geom. 44(2), 121–127 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM J. Comput. 44(6), 1626–1649 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bose, P., Morin, P.: Online routing in triangulations. SIAM J. Comput. 33(4), 937–951 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: SODA, pp. 291–300 (1993)Google Scholar
  6. 6.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42(1), 67–90 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chew, P.: There is a planar graph almost as good as the complete graph. In: SOCG, pp. 169–177 (1986)Google Scholar
  8. 8.
    Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5, 399–407 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P.: Routing in unit disk graphs. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) LATIN 2016. LNCS, vol. 9644, pp. 536–548. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49529-2_40 CrossRefGoogle Scholar
  10. 10.
    Keil, J.M., Gutwin, C.A.: Classes of graphs which approximate the complete euclidean graph. Discrete Comput. Geom. 7, 13–28 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Misra, S., Woungang, I., Misra, S.C.: Guide to Wireless Sensor Networks, 1st edn. Springer Publishing Company, Incorporated (2009)Google Scholar
  12. 12.
    Narasimhan, G., Smid, M.H.M.: Geometric spanner networks. Cambridge University Press (2007)Google Scholar
  13. 13.
    Xia, G.: The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput. 42(4), 1620–1659 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Jean-Lou De Carufel
    • 2
  • Vida Dujmović
    • 2
  • Frédérik Paradis
    • 2
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada

Personalised recommendations