This paper addresses the basic question of how well can a tree approximate
distances of a metric space or a graph. Given a graph, the problem of
constructing a spanning tree in a graph which strongly preserves distances in
the graph is a fundamental problem in network design. We present scaling
distortion embeddings where the distortion scales as a function of Ο΅,
with the guarantee that for each Ο΅ the distortion of a fraction
1βΟ΅ of all pairs is bounded accordingly. Such a bound implies, in
particular, that the \emph{average distortion} and βqβ-distortions are
small. Specifically, our embeddings have \emph{constant} average distortion and
O(lognβ)β2β-distortion. This follows from the following
results: we prove that any metric space embeds into an ultrametric with scaling
distortion O(1/Ο΅β). For the graph setting we prove that any
weighted graph contains a spanning tree with scaling distortion
O(1/Ο΅β). These bounds are tight even for embedding in arbitrary
trees.
For probabilistic embedding into spanning trees we prove a scaling distortion
of O~(log2(1/Ο΅)), which implies \emph{constant}
βqβ-distortion for every fixed q<β.Comment: Extended abstrat apears in SODA 200