# 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

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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