$\log_{2}{(\frac{1}{2}n)} + \log_{2}{(\frac{1}{4}n}) + \log_{2}{(\frac{1}{8}n)} + \ldots + \log_{2}{(\frac{1}{2^{log_{2}(n)}}n)} = O(\log_{2}(n))$?

Question

I am trying to analyze a series that I found, in the analysis of an algorithm. And I was wondering if the following was true: $$\log_{2}{\left(\frac{1}{2}n\right)} + \log_{2}{\left(\frac{1}{4}n\right)} + \log_{2}{\left(\frac{1}{8}n\right)} + \cdots + \log_{2}{\left(\frac{1}{2^{log_{2}(n)}}n\right)} = \mathcal{O}\left(\log_{2}(n)\right)$$

It is basically the sum of the geometric series which converges to $1$, but each term is multiplied by $n$ and $\log_{2}$ is taken.

Using the logarithm rule $\log_{2}(a) + \log_{2}(b) = \log_{2}(ab)$ seems to give results that indicate this is not the case right? e.g. $$\log_{2}(128) = 7$$ and $$ \log_{2}\left(\frac{1}{2}128\right) + \log_{2}\left(\frac{1}{4}128\right) \ldots = \log_{2}(64) + \log_{2}(32) + \log_{2}(16) + \log_{2}(8) + \log_{2}(4) + \log_{2}(2) = \log_{2}(2097152) = 21 $$ So a difference of 14, when $n=128$, I suppose that there is no way that this series is upper bounded by like $\mathcal{O}(10\log_{2}(n))$.

Answer 1

Using $\log a + \log b = \log (ab)$, we get: $$\sum\limits_{k=1}^{\log_2 n}\log_2\left(\frac1{2^k}n\right) = \log_2\left(\prod\limits_{k=1}^{\log_2 n}\frac{n}{2^k} \right) = \log_2\left(\frac{n^{\log_2 n}}{2^{\sum\limits_{k=1}^{\log_2 n}k}}\right) = \log_2\left(\frac{n^{\log_2 n}}{2^{\frac{\log_2 n\times(\log_2 n + 1)}{2}}}\right)$$

But: $$\frac{n^{\log_2 n}}{2^{\frac{\log_2 n\times(\log_2 n + 1)}{2}}} = \frac{n^{\log_2 n}}{n^{\frac{\log_2 n + 1}{2}}} = n^{\frac{\log_2 n - 1}{2}}$$

We conclude that the sum is equal to: $$\log_2 \left(n^{\frac{\log_2 n - 1}{2}}\right) = \frac{\log_2 n - 1}{2}\log_2 n \in \Omega((\log n)^2)$$

So the answer is no.

Another way to get this result is to say that half (so $\frac12\log_2 n$) of the terms of the sum are greater than $\log_2(\sqrt{n})$.

Answer 2

Let $k:=\log_2 n$ then

$\sum\limits_{j=1}^{\log_2 n}\log_2(\frac{1}{2^j} \cdot n) = \sum\limits_{j=1}^{k}(\log_2(\frac{2^k}{2^j})) = \sum\limits_{j=1}^{k}\log_2(2^j) = \sum\limits_{j=1}^{k} j = \frac 1 2 k(k+1)$

so

$\sum\limits_{j=1}^{\log_2 n}\log_2(\frac{1}{2^j} \cdot n) \in \Theta(k^2) =\Theta((\log_2 n)^2)$

Answer 3

Yet another way to obtain the bound using the identity $\log \frac ab=\log a - \log b$ to have simpler sums.

$ \sum\limits_{i=1}^{\log_2 n} \log_2 \left({\frac{n}{2^i}}\right) = \sum\limits_{i=1}^{\log_2 n} \left(\log_2 n - \log_2 2^i \right) = \sum\limits_{i=1}^{\log_2 n} \left(\log_2 n - i \right) = \sum\limits_{i=1}^{\log_2 n} \log_2 n - \sum\limits_{i=1}^{\log_2 n} i $

We now have,

$ \sum\limits_{i=1}^{\log_2 n} \log_2 n - \sum\limits_{i=1}^{\log_2 n} i = (\log_2 n)^2 - \left(\frac {\left(\log_2n\right)^2}{2} + \frac{\log_2 n}{2}\right) = \frac {\left(\log_2n\right)^2}{2} - \frac{\log_2 n}{2} = \Theta\left( \left(\log_2 n\right)^2 \right) $