Discrete & Computational Geometry

, Volume 29, Issue 1, pp 77-81

First online:

A Lower Bound on the Distortion of Embedding Planar Metrics into Euclidean Space

  • Ilan NewmanAffiliated withComputer Science Department, University of Haifa, Haifa 31905, Israel ilan@cs.haifa.ac.il, yuri@cs.haifa.ac.il
  • , Yuri RabinovichAffiliated withComputer Science Department, University of Haifa, Haifa 31905, Israel ilan@cs.haifa.ac.il, yuri@cs.haifa.ac.il

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Abstract.

We exhibit a simple infinite family of series-parallel graphs that cannot be metrically embedded into Euclidean space with distortion smaller than
$$\Omega(\sqrt{\log n})$$
. This matches Rao's [14] general upper bound for metric embedding of planar graphs into Euclidean space, thus resolving the question how well do planar metrics embed in Euclidean spaces?