Recently, I started (independent) learning of the theory of metric embeddings from the Fall 2003 course offered at CMU .
I had a very basic question from the very first lecture of this course which I would like to get more intuition about. On page, $5$, the notes say that this technique can be used in a straightforward way to give an $\alpha$ approximation algorithms for problems like TSP if (say) the following hold.
(i) The metric embeds into a tree.
(ii) The embedding has distortion at most $\alpha$
What I am not sure about is whether the solution generated by using the embedding is even valid - because for all I know, it could be that the TSP solution on the tree uses only from among those edges which were contracted.
To be more precise, I feel more comfortable accepting that if we have got a mapping $f$ from the original space $(X,d)$ to the tree metric $(V, d')$ which expands all the pairwise distances, then I can use the TSP solution on this tree as an approximate solution to the TSP problem on the original metric with approximation factor same as the expansion of the mapping $f$. I am not sure about how approximation factor can be the same as (or even related to) the distortion of the mapping $f$.