Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier

Question

Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.

Q1. What is the least time complexity possible/known for recreating the Integral point set $P$ given $D$ through a Turing machine computable algorithm?

Q2. What is the shortest string which can describe $P$?

For example:

• Using $D$ to describe $P$ may be a shorter method than $P$ itself;

• Scaling $D$ down to elements all relatively prime may still shorten the description string.

• Using a certain set of computed values to describe $P$ may further minimize the expression ( provided they exist).

• Just $P= \text{IntegralPointSet}(n)$ will be problematic as

  • It is not Turing computable. (Comments?)
  • It does not define exactly the point set we want out of possibly many such integral point sets.

Which gets me to the third question:

Q3. What is the shortest string required to uniquely identify a particular integral point set of cardinality $n$ amongst all other such integral point sets?

Answer 1

This is not a direct answer, but rather an historical, and somewhat tangential comment. Back in 1979, the general problem you posed was proved NP-hard by Saxe and by Yemini independently. There has been quite a rich literature on this topic in the last 35+ years, which you might trace via Google Scholar. Added. For example, see the 2014 survey below.

James B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Allerton Conference on Communications, Control, and Computing, pp. 480–489, 1979. Also in James B. Saxe: Two Papers on Graph Embedding Problems, Department of Computer Science, Carnegie-Mellon University, 1980. (PDF download.)

Yechiam Yemini. Some theoretical aspects of position-location problems. In 20th Annual Symposium on Foundations of Computer Science (FOCS), pp. 1–8, Oct. 1979. DOI: 10.1109/SFCS.1979.39 (ACM link.)

Liberti, Leo, Carlile Lavor, Nelson Maculan, and Antonio Mucherino. "Euclidean distance geometry and applications." Siam Review 56(1) (2014): 3-69. (Journal link.)

          
           _{(Figure from "Untangling planar graphs from a specified vertex position—Hard cases" (Elsevier link).)}