# Solving T(n) = 2T(n/2) + log n with the recurrence tree method

I was solving recurrence relations. The first recurrence relation was

$T(n)=2T(n/2)+n$

The solution of this one can be found by Master Theorem or the recurrence tree method. The recurrence tree would be something like this:

The solution would be:

$T(n)=\underbrace{n+n+n+...+n}_{\log_2{n}=k \text{ times}}=\Theta(n \log{n})$

Next I faced following problem:

$T(n)=2T(n/2)+\log{n}$

My book shows that by the master theorem or even by some substitution approach, this recurrence has the solution $\Theta(n)$. It has same structure as above tree with the only difference that at each call it performs $\log{n}$ work. However I am not able to use same above approach to this problem.

The non-recursive term of the recurrence relation is the work to merge solutions of subproblems. The level $k$ of your (binary) recurrence tree contains $2^k$ subproblems with size $\frac {n}{2^k}$, so you need at first to find the total work on the level $k$ and then to summarize this work over all the tree levels.

For example, if the work is constant $C$, then the total work on the level $k$ will be $2^k \cdot C$, and the total time $T(n)$ will be given by the following sum:

$$T(n) = \sum_{k=1}^{\log_2{n}}2^k C = C(2^{\log_2{n}+1}-2) = \Theta(n)$$

However, if the work logarithmically grows with the problem size you'll need to accurately compute the solution. The series would be like the following:

$$T(n)=\log_2{n}+\underbrace{2\log_2(\frac {n} {2})+4\log_2(\frac {n} {4})+8\log_2(\frac {n} {8}) + ....}_{\log_2{n} \text{ times}}$$

It'll be a quite complex sum:

$$T(n)=\log_2{n}+\sum_{k=1}^{\log_2{n}}2^k \log_2(\frac {n} {2^k})$$

I'll temporarily denote $m=\log_2{n}$ and simplify the summation above:

$$\sum_{k=1}^{m}2^k \log_2(\frac {n} {2^k})=\\=\sum_{k=1}^{m}2^k(\log_2{n}-k)=\\=\log_2{n}\sum_{k=1}^{m}2^k-\sum_{k=1}^{m}k2^k=\\=\log_2{n}(2^{m+1}-2)-(m2^{m+1}-2^{m+1}+2)$$

Here I used a formula for the $\sum_{k=1}^{m}k2^k$ sum from the Wolfram|Alpha online symbolic algebra calculator. Then we need to replace $m$ back by $\log_2{n}$:

$$T(n)=\log_2{n}+2n\log_2{n}-2\log_2{n}-2n\log_2{n}+2n-2$$ $$=2n-\log_2{n}-2=\Theta(n)$$

Q.E.D.

• damn I made super stupid mistake in asking question. Now corrected. However now am not able to get how the stuff cancel each other in the series: $2^0\log_2\frac{n}{2^0}+2^1\log_2\frac{n}{2^1}+2^2\log_2\frac{n}{2^2}+...+2^{ \log_2{n}}\log_2\frac{n}{2^{ \log_2{n}}}=\log_2{n}+2\log_2{\frac{n}{2}}+4\log_2{\frac{n}{4}}+...+n\log_2{1}$.. This is the series that you come up with right? Trying with $n=8$, I got $\log_2{8}+2\log_2{4}+4\log_2{2}+8\log_2{1}=3+4+4=11$. Whats wrong here? – anir May 14 '16 at 12:05
• @anir - Use equality $\log_2(\frac{n}{2^k}) = \log_2 n - k$ – HEKTO May 14 '16 at 15:36
• I still didn't get how that series collapses to $\Theta(n)$ :'¬( Math-noob-here... – anir May 19 '16 at 18:42
• @anir - I'll expand my answer – HEKTO May 19 '16 at 22:43
• @HEKTO If you solve above equ of comment, you still get nlog(n) ?? I have tried a lot. would you please help me here ? – roottraveller Jul 15 '17 at 17:15