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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?

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up vote 1 down vote accepted

This is not a direct answer, but rather an historical, and somewhat tangential comment. Way 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 30+ years, which you might trace via Google Scholar.

[Sax79] 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)

[Yem79] 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)

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

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