# Construction of an integral point set given the set of distances ,its minimal description so as 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?

  eg
-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 minimise
the expression ( provided they exist).

-Just P= 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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@ARi, what do you mean by "integral point set"? –  Włodzimierz Holsztyński Jun 19 '13 at 19:18
@ARi, thank you. –  Włodzimierz Holsztyński Jun 21 '13 at 20:39