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.
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Cao, L., Strelzoff, A., Sun, J.Z. (2010). A Lower Bound on Greedy Embedding in Euclidean Plane. In: Bellavista, P., Chang, RS., Chao, HC., Lin, SF., Sloot, P.M.A. (eds) Advances in Grid and Pervasive Computing. GPC 2010. Lecture Notes in Computer Science, vol 6104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13067-0_25
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DOI: https://doi.org/10.1007/978-3-642-13067-0_25
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