Improved embeddings of graph metrics into random trees

Over the past decade, numerous algorithms have been developed using the fact that the distances in any <i>n</i>-point metric (<i>V, d</i>) can be approximated to within <i>O</i>(log <i>n</i>) by distributions <i>D</i> over trees on the point set <i>V</i> [3, 10]. However, when the metric (<i>V, d</i>) is the shortest-path metric of an edge weighted graph <i>G</i> = (<i>V, E</i>), a natural requirement is to obtain such a result where the support of the distribution <i>D</i> is only over <i>subtrees</i> of <i>G</i>. For a long time, the best result satisfying this stronger requirement was a exp {√log <i>n</i> log log <i>n</i>} distortion result of Alon et al. [1]. In a recent breakthrough, Elkin et al. [9] improved the distortion to <i>O</i>(log² <i>n</i> log log <i>n</i>). (The best lower bound on the distortion is Ω(log <i>n</i>), say, for the <i>n</i>-vertex grid [1].)In this paper, we give a construction that improves the distortion to <i>O</i>(log² <i>n</i>), improving slightly on the EEST construction. The main contribution of this paper is in the analysis: we use an algorithm which is similar to one used by EEST to give a distortion of <i>O</i>(log³ <i>n</i>), but using a new probabilistic analysis, we eliminate one of the logarithmic factors. The ideas and techniques we use to obtain this logarithmic improvement seem orthogonal to those used earlier in such situations---e.g., Seymour's decomposition scheme [4, 9] or the cutting procedures of CKR/FRT [5, 10], both which do not seem to give a guarantee of better than <i>O</i>(log² <i>n</i> log log <i>n</i>) for this problem. We hope that our ideas (perhaps in conjunction with some of these others) will ultimately lead to an <i>O</i>(log <i>n</i>) distortion embedding of graph metrics into distributions over their spanning trees.