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 Θ( √ n) distortion in the worst case [19, 1], 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 Matouˇsek [21] by proving that some planar graph metrics require Ω(n2/3 ) 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 Ω(n) distortion in any crossingfree straight-line ...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, Erik D. Demaine, Mohammad Moharrami
• Keywords: COMPGEOM 2006, Graph Metrics, Planar Graph, Planar Graph Metrics