We connect the results of Bartal 4] on probabilistic approximation of metric spaces by tree metrics, and Klein, Plotkin and Rao 11] on decompositions of graphs without small K s;s minors (such as planar graphs) to show that metrics induced by such graphs can be probabilistically approximated by tree metrics with an O(log d) distortion, where d is the diameter of the given graph. This improves upon Bartal's result that holds for general n-node metrics with a distortion of O(log n log log n). The main ingredient of our proof is that weak probabilistic partitions su ce for the construction of tree metrics with low distortion, in contrast to strong partitions used by Bartal. We also show that our result is the best possible by providing a lower bound of (log d) for the distortion of any probabilistic approximation of the square grid by tree metrics.
Konjevod et al. REF showed that for any positive integer s, the metric of G without a K s,s minor can be probabilistically approximated by a special case of tree metric, called r-hierarchically well separated tree (r-HST) with distortion O(log diam G).
On Approximating Planar Metrics by Tree Metrics
Information Processing Letters
Mathematics Computer Science