The problem can be describe in the following way:

Given a positive integer $N$, we need to find a methods to generate a set of integers A, so that for any integer $1 \leq k \leq N$, we can find two integers $x, y \in A$, such that $x - y = k$.

An example for problem is the following: $N=9$, we can generate a set $A:=\{-3, -2, -1, 0, 3, 6\}$. Clearly, the size of $A=6$.

We naively prove that the minimal size of $|A| = \sqrt(2N)$. However, we can't find a method to generate an $A$ that can reach this theoretical minimal. The best methods we find can only reach $|A| = 2 \sqrt(N)$.

Is there any methods that can reach the theoretical minimal $\sqrt(2N)$? If not, what is the minimal size that any methods can reach?