Find a maximum of difference without sorting. Suppose that I have an unsorted array of real numbers $\{a_i\}_{i=1}^n$. We sorted it in ascending order and now we have array $\{b_i\}_{i=1}^n$. Let $N=\max\limits_{i=\overline{1..n-1}} |b_{i+1}-b_{i}|$
I want to know $N$, but if I will sort the array $\{a_i\}_{i=1}^n$ and then will find $N$, the complexity of all operation will be $\mathcal{O}\left(n\log n \right)$. It isn't fast enough. 
Is it possible to find better way, without sorting?

In the Algebraic Decision Tree model, we can prove an $\Omega(n\log n)$ lower bound for your problem.
This follows because the Element Distinctness Problem has the same lower bound and can be reduced to your problem.
