I have a collection of points with pairwise-distances between points that is not Euclidean (they are actually shortest path distances, involving travel thru many other nodes which are not of further interest in the problem). A small example would be:
$O = (0, 0)$, $O\to A = 11$, $O\to B = 12$, $O\to C = 13$
$A = (1, 2)$, $A\to B = 21$, $A\to C = 22$
$B = (3, 4)$, $B\to C = 31$
$C = (5, 6)$
It can be seen as an undirected graph (so $X\to Y == Y\to X$ for all $X,Y$ in $\{O, A, B, C\}$).
What I would like to be able to do is transform it into a Euclidean space such that the distances between points are still correct. It does not matter at what point any of the nodes are placed at, only the distance between matters. The reason is to be able to make use of the triangle inequality and relatively simpler geometry to solve a vehicle routing problem.
As another way of explaining that may be more helpful, imagine you have a map with several cities on it and also the roads connecting those cities that form the shortest route between each city and all others. So the distance betwen city A and B is going to be greater than the Euclidean distance in general. I would like to take that map and stretch it non-uniformly, so at the end the straight line distance between each pair of cities is the shortest path distance from the original map. Is there some sort of process or algorithm to do this? Or perhaps this is actually impossible? Any advice on it really appreciated.
The closest question I could find to mine is Given distances (shortest paths) between four cities, how to show that they cannot be in the same plane?, but I cannot see how this might be applied to my problem.
EDIT (following Alex Ravsky's comment with link to a related question/answer)
According to the linked to post, for what I want to do to be possible, the square distance matrix below would need to be of conditionally negative type?
$[[0, 11, 12, 13],$
$ [11, 0, 21, 22],$
$ [12, 21, 0, 31],$
$ [13, 22, 31, 0]]$
I have no idea if my actual distance matrix is conditionally negative (I can find out of course) but I also need to know HOW to transform/embed my space into the Euclidean plane, assuming it is of the right form.
