On a Conjecture Related to Geometric Routing

  • Christos H. Papadimitriou
  • David Ratajczak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3121)

Abstract

We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a subgraph, 2-dimensional virtual coordinates for the nodes can be found for which greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms. We show a weaker result, namely that for any network containing a 3-connected planar subgraph, 3-dimensional virtual coordinates always exist enabling a form of greedy routing inspired by the simplex method; we provide experimental evidence that this scheme is quite effective in practice. We also propose a rigorous form of face routing based on the Koebe-Andre’ev-Thurston theorem. Finally, we show a result delimiting the applicability of our approach: any 3-connected K3,3-free graph has a planar 3-connected subgraph.

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References

  1. 1.
    Kuhn, F., Wattenhofer, R., Zhang, Y., Zollinger, A.: Geometric Ad-Hoc Routing: Of Theory and Practice. In: Proc. 22nd ACM Int. Symposium on the Principles of Distributed Computing, PODC (2003)Google Scholar
  2. 2.
    Karp, B., Kung, H.T.: GPSR: greedy perimeter stateless routing for wireless networks. In: Mobile Computing and Networking, pp. 243–254 (2000)Google Scholar
  3. 3.
    Rao, A., Papadimitriou, C., Shenker, S., Stoica, I.: Geographic routing without location information. In: Proceedings of the 9th annual international conference on Mobile computing and networking, pp. 96–108. ACM Press, New York (2003)CrossRefGoogle Scholar
  4. 4.
    Tutte, W.T.: Convex representations of graphs. Proceedings London Math. Society 10, 304–320 (1960)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Linial, N., Lovasz, L., Wigderson, A.: Rubber bands, convex embeddings and graph connectivity. Combinatorica 8, 91–102 (1988)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ziegler, G.M.: Lectures on Polytopes. Springer, Berlin (1995)MATHGoogle Scholar
  7. 7.
    Thurston, W.: Three-dimensional Geometry and Topology. Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)MATHGoogle Scholar
  8. 8.
    Lovasz, L.: Steinitz representations of polyhedra and the colin de verdiere number. Journal of Combinatorial Theory B, 223–236 (2001)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Christos H. Papadimitriou
    • 1
  • David Ratajczak
    • 1
  1. 1.Computer Science DivisionUniversity of California at BerkeleyBerkeleyUSA

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