A tight bound on approximating arbitrary metrics by tree metrics

Abstract

In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

Resilient and Low Stretch Routing through Embedding into Tree Metrics

“…The leaves of T are the vertices in P , and the internal nodes are Steiner nodes. Fakcharoenphol, Rao and Talwar [11] have shown that for any metric (P, d) one can find a family of trees such that a randomly selected metric from the family has expected distortion of O(log n), which is also tight.…”

“…Review of The FRT Algorithm [11]. Without loss of generality, we assume that the smallest distance between any two vertices in P is 1 and the diameter of P is ∆.…”

“…In this case we have their distance on the tree to be d H (u, v) = 2 ∑ i j=1 2 j = 2 i+2 . Fakcharoenphol, Rao and Talwar [11] (u, v), in expectation over all random choices of β and π. A distributed implementation of the algorithm is available in [12].…”

“…Constant Distortion Embedding in Two HSTs For a given metric (P, d) with expansion rate k, we build two HSTs, H 1 and H 2 with independent, random seeds using the algorithm in [11]. For any two points u, v in P , we define the distance between them to be the minimum shortest path in the two trees.…”

“…Approximating a metric with probabilistic hierarchical well-separated trees was first proposed by Bartal [6,5], with the motivation that many problems are easier to solve on a tree than on a general graph. Later, Fakcharoenphol et al [11] improved the distortion to O(log n) for any n node metric and this is tight.…”