Orders on metric spaces and invariants

Let $M$ be a metric space and let $T$ be a total ordering of its points.

For a finite subset $X\subset M$ we calculate the minimal length $l_{opt}(X)$ of a path visiting all its points, the length $l_T(X)$ of the path which visits the points of $X$ with respect to the order $T$, and the ratio $r_T(X) = l_T(X)/l_{opt}(X)$.

As J. Bartholdi and J.Platzman noticed in 1982, for the square $[0;1]^2$ and the order of the first visit of the Peano curve all such ratios $r_T(X)$ are bounded by a logarithmic function of $|X|$.

For a given metric space $M$ we want to find an order with as small such bound on ratios as possible.

We connect existence of "good" orders with more traditional properties and invariants of metric spaces, such as hyperbolicity, Assouad-Nagata dimension, number of ends and doubling.