, Volume 29, Issue 1, pp 77-81

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

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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?