The first article above concerns finding information about a finite set of given distances between them.
I think in the last one, one considers distances measured with random error, and there is uncertainty in ones estimate of the dimension of the Euclidean space in which the points would fit if they had been measured exactly.
Notice that for some finite metric spaces, there is no Euclidean space in which they can be embedded. For example:
d(A,B) & = 1 \\
d(A,C) & = 1 \\
d(A,D) & = 1 \\
d(B,C) & = 1 \\
d(C,D) & = 1 \\
d(B,D) & = 2
This puts $B$, $C$, and $D$ on a common straight line, and makes $A$ equidistant from all three. Draw the picture.