I am currently researching a specific topic and i stumbled upon a problem, for which i cannot find enough information. I will be really thankful, if you can share your knowledge or resources, in order for me to get a better grip of the problem.
I have a graph given as 2D a weighted adjacency matrix, where weights represent Euclidic distance between the vertices. My question is the following, is it possible to represent the edge distance accurately on a 2D grid and what computational time it would take (assuming that the scale of the plot is allowing all the distances to be represented). The vertex positioning, in relation to the (x,y) coordinates of the vertices is not given, so it may vary in order for the distances to be accurately represented.
I am rather interested in the algorithmic approach behind this problem, than the code done with networkx in python or similar.
The idea i have is to start with the first vertex, position the 2nd accordingly (considering the distance to the 1st) then position the 3rd accordingly, (considering the distance to 1st and 2nd) and so on. However i am neither sure if this is optimal, nor sure of it is computational efficiency (altough i think it is $\in O(n^2)$) for the 2D case, when not considering the calculation of the intersection of all distances. What one could do when distances get bigger in number is to draw discs with the distance from a certain already drawn point to the next, and then look at the intersection section of all discs .(?) I am not sure however, that it will always be possible for all discs to interesect (?).
Any help is appreciated, thanks in advance !
Edit: I found this topic which refers to the problem as a subproblem of distance geometry. There is also a given paper, which refers, that for the K dimensional case the problem is NP-hard. However i have still not found any additional information for the 2D case.