A lower bound on greedy embedding in euclidean plane

  • Authors:

  • Lei Cao;Andrew Strelzoff;Jonathan Z. Sun

  • Affiliations:

  • Department of Electrical Engineering, University of Mississippi University, MS;School of Computing, University of Southern Mississippi, Hattiesburg, MS;School of Computing, University of Southern Mississippi, Hattiesburg, MS

  • Venue:
  • GPC'10 Proceedings of the 5th international conference on Advances in Grid and Pervasive Computing
  • Year:
  • 2010

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Abstract

Greedy embedding is the key of geometric greedy routing in p2p networks It embeds a graph (the topological structure of the p2p network) in a metric space such that between any source-destination pair, there is a distance decreasing path for message delivery It is known that any graph can be greedily embedded in the hyperbolic plane with using O(logn) bits for each node's coordinates [7] Interestingly, on greedy embedding in the Euclidean plane, existing embedding algorithms result in coordinates with ${\it \Omega}(n)$ bits It was recently proved that ${\it \Omega}(n)$ is a lower bound of the coordinates' bit length if one uses Cartesian or polar coordinate system and preserves the planar embedding of a planar graph when greedily embedding it in the Euclidean plan [2] In this paper we strengthen this result by further proving that ${\it \Omega}(n)$ is still a lower bound even if the graph is allowed to take free embedding in the plane.