Plane embeddings of planar graph metrics

Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Theta(root n) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matousek [M3] by proving that some planar graph metrics require Omega(n(2/3)) distortion in any embedding into the plane, proving the first separation between these two types of graph metrics. We also prove that some planar graph metrics require Q(n) distortion in any crossing-free straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied notion of crossing-free straight-line planar drawings. Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with O(root n) distortion, generalizing the previous results on trees (both the worst-case bound and the approximation algorithm) and building techniques for handling cycles in plane embeddings of graph metrics.