What is wrong with this solution for $\mathcal{O}({\log({n \choose \frac{n}{2}})})$?

Question

In this recitation on MIT OCW, the instructor uses Stirling's approximation to calculate that

$\mathcal{O}({\log({n \choose \frac{n}{2}})}) = \mathcal{O}(n)$.

However, I went through the following steps to conclude that $\mathcal{O}({\log({n \choose \frac{n}{2}})}) = \mathcal{O}(\log{n})$. Where did I go wrong?

First, note that ${n \choose \frac{n}{2}} = \frac{n!}{\frac{n}{2}!\frac{n}{2}!}$. By basic logarithm laws, we get that this is equal to $\log{(n!)} - \log{(\frac{n}{2}!\frac{n}{2}!)}$. From this it follows that:

$$ \mathcal{O}(\log{(n!)} - \log{(\frac{n}{2}!\frac{n}{2}!)})\\ = \mathcal{O}(\log{(n!)}) \\ = \mathcal{O}\Big(\log{\big(n(n-1)(n-2)\cdots(1)\big)}\Big) \\ = \mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big) = \mathcal{O}(\log{n}) $$

So, what is wrong here? I've gone over it for a while and I can't see any mistakes. Plotting these functions, though, I can see quite clearly that there must be a mistake.

Answer 1

$\mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big) = \mathcal{O}(\log{n})$

That is not right. When $n$ is large enough,

$$\begin{align} \log{n} + \log{(n-1)} + \ldots + \log(1) &\ge \log{n} + \log{(n-1)} + \ldots + \log(n/2)\\ &\ge n/2 \log(n/2)\\ &=\Theta(n\log n) \end{align}$$

More precisely, since $n!\sim {\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}$ by Stirling's approximation,

$$\begin{align} \log{n} + \log{(n-1)} + \ldots + \log(1) &=\log(n!)\\ &\sim \log\left({\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}\right) \\ &\sim n(\log(n) -1) \\ &\sim n\log(n) \end{align}$$

You can take a moment to try understanding intuitively why $$\lim_{n\to\infty}\frac {\log(n!)}{\log (n^n)} = 1.$$

Answer 2

The last step is incorrect. $\mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big)$ is not $\mathcal{O}(\log{n})$. You made the same mistake as What goes wrong with sums of Landau terms?. See also What is the asymptotic runtime of this nested loop?.