TITLE: Feasibility of a list of prescribed distances in R^3
QUESTION [8 upvotes]: I am puzzled with the following problem:
Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every of the given numbers represents a shortest distance which belongs to a distinct pair of points?"
What is an efficient algorithm to solve such problem? If I understand properly, first I have to find $m$ from $n=\frac{m(m+1)}{2}$ which is the number of such points. But what is the next step? Should I deal with checking the triangle inequality (which seems to be very inefficient) or what?
Thank you in advance!

REPLY [11 votes]: What did not seem to be mentioned in this old thread 
is that this is a well-studied problem that goes under the name 
distance geometry, which Wikipedia defines as "the characterization and study of sets of points based only on given values of the distances between member pairs."
The problem has applications especially to molecular structure.  Crippen and Havel published a
book on this in 1988: Distance geometry and molecular conformation (Research Studies Press, Taunton, Somerset, England), with an update by Crippen later ("Chemical distance geometry: Current realization and future projection").  There are many papers on the topic, published in venues like the Journal of Computational Chemistry, e.g., "Molecular conformations from distance matrices" (Volume 14, Issue 1, pages 114–120, January 1993).  Because of the many applications, there are software packages
developed to solve the problem; the Wikipedia article has a list.
The problem is computationally intractable; the proof that one version is NP-hard goes back
to a 1979 paper by Saxe, "Embeddability of weighted graphs in $k$-space is strongly NP-hard."