  We introduce irreducible correspondences that enables us to calculate the
Gromov--Hausdorff distances effectively. By means of these correspondences, we
show that the set of all metric spaces each consisting of no more than $3$
points is isometric to a polyhedral cone in the space $R^3$ endowed with the
maximum norm. We prove that for any $3$-point metric space such that all the
triangle inequalities are strict in it, there exists a neighborhood such that
the Steiner minimal trees (in Gromov-Hausdorff space) with boundaries from this
neighborhood are minimal fillings, i.e., it is impossible to decrease the
lengths of these trees by isometrically embedding their boundaries into any
other ambient metric space. On the other hand, we construct an example of
$3$-point boundary whose points are $3$-point metric spaces such that its
Steiner minimal tree in the Gromov-Hausdorff space is not a minimal filling.
The latter proves that the Steiner subratio of the Gromov-Hausdorff space is
less than 1. The irreducible correspondences enabled us to create a quick
algorithm for calculating the Gromov-Hausdorff distance between finite metric
spaces. We carried out a numerical experiment and obtained more precise upper
estimate on the Steiner subratio: we have shown that it is less than $0.857$.
