Since it's for class, I won't actually build the DFA for you, but I will give you a way to look at it that'll help. As far as $b$s go, you really only care about whether or not you've seen a $b$ so ...

The statement: If $f(n)=O(g(n))$ then for large $n$ , $f(n)$ would grow as fast as $g(n)$. is incorrect, and that's probably where your confusion is. Taking from Raphael's answer in How does one ...

Simply put, this statement of yours: On the other hand $\tilde{L}$ seems to contain strings E does not, i.e. those that are formed by concatenating non-empty string from both languages. is wrong....

What the article means by $$f : \{0,1\}^n \rightarrow \{0,1\}^n$$ is that the input and output of the function is a binary string of length $n$. As for the problem itself, the idea is that we have ...

You probably misunderstood the question. For starters, the pumping lemma can't be used to prove that a language is regular, it can only be used to prove that a language isn't regular. The Wikipedia ...

I can get you an improvement for the first step; every time you do that first step, you're basically trying to find the minimum, and then continue on the rest of the array. You don't need to do all ...