How to solve this recurrence $T(n)=2T(n/2)+n/\log n$

Question

How can I solve the recurrence relation $$T(n)=2T\left(\frac n2\right)+\frac{n}{\log n}$$?

I am stuck up after few steps..

I arrive till

$$T(n) = 2^k T(1) + \sum_{i=0}^{\log(n-1)} \left(\frac{n}{\log n} - i\right)$$

How to simplify this log summation?

Answer 1

$$S(k)=2^{-k}\cdot T(2^k)\implies S(k)=S(k-1)+\frac1{k\log2} $$ $$ S(k)=S(0)+\frac{H_k}{\log2}\implies T(2^k)=\Theta(2^k\cdot\log k)$$ ...Which does not imply that $T(n)=\Theta(n\cdot\log\log n)$, although this might be the conclusion suggested in your textbook.

Answer 2

Suppose you have $T(0) = 0$ and $T(1) = 1$ and your recurrence for $n\ge 2$ is $$T(n) = 2 T(\lfloor n/2 \rfloor) + \frac{n}{\lfloor\log_2 n\rfloor}.$$ This gives the following exact formula for $T(n)$ where $n\ge 2:$ $$T(n) = 2^{\lfloor \log_2 n \rfloor} + \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{1}{\lfloor \log_2 n \rfloor - j} \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^k$$ where we suppose that the binary representation of $n$ is $$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k.$$ The reader is invited to verify this formula which is not at all difficult.

Now for an upper bound consider a string of one digits which gives $$T(n) \le 2^{\lfloor \log_2 n \rfloor} + \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{1}{\lfloor \log_2 n \rfloor - j} \sum_{k=j}^{\lfloor \log_2 n \rfloor} 2^k \\ = 2^{\lfloor \log_2 n \rfloor} + \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{1}{\lfloor \log_2 n \rfloor - j} (2^{\lfloor \log_2 n \rfloor+1} - 2^j) \\= 2^{\lfloor \log_2 n \rfloor} + 2^{\lfloor \log_2 n \rfloor+1} \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{1}{\lfloor \log_2 n \rfloor - j} - \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{2^j}{\lfloor \log_2 n \rfloor - j} \\= 2^{\lfloor \log_2 n \rfloor} + 2^{\lfloor \log_2 n \rfloor+1} H_{\lfloor \log_2 n \rfloor} - \sum_{j=1}^{\lfloor \log_2 n \rfloor} \frac{2^{\lfloor \log_2 n \rfloor - j}}{j} \\= 2^{\lfloor \log_2 n \rfloor} + 2^{\lfloor \log_2 n \rfloor+1} H_{\lfloor \log_2 n \rfloor} - 2^{\lfloor \log_2 n \rfloor} \sum_{j=1}^{\lfloor \log_2 n \rfloor} \frac{2^{- j}}{j}.$$ Observe that the remaining sum term converges to a number, so that we get the following asymptotics for the upper bound: $$2^{\lfloor \log_2 n \rfloor} (1+2H_{\lfloor \log_2 n \rfloor} -\log 2).$$ This upper bound is actually attained and hence cannot be improved upon, just like the lower bound, which we now compute and which occurs for a one digit followed by a string of zero digits, giving $$ T(n) \ge 2^{\lfloor \log_2 n \rfloor} + \sum_{j=0}^{\lfloor \log_2 n \rfloor - 1} \frac{1}{\lfloor \log_2 n \rfloor - j} 2^{\lfloor \log_2 n \rfloor} = 2^{\lfloor \log_2 n \rfloor} (1+H_{\lfloor \log_2 n \rfloor}).$$ Joining the dominant terms of the upper and the lower bound we get a complexity of $$\Theta\left(2^{\lfloor \log_2 n \rfloor} \times H_{\lfloor \log_2 n \rfloor}\right) = \Theta\left(2^{\log_2 n} \log \log n\right)= \Theta\left(n\log \log n\right).$$

This MSE link points to a chain of similar computations.

Answer 3

Your summation is wrong, and you should have replaced $k$ with $log(n)$ in your last expression.

Here are detailed steps of this recurrence relation:

$$ \\ T(n) = 2T(\frac{n}{2}) + \frac{n}{\log n} \\ T(n) = 2(2T(\frac{n}{4}) + \frac{\frac{n}{2}}{\log \frac{n}{2}}) + \frac{n}{\log n} = 2^2T(\frac{n}{2^2}) + \frac{n}{\log(n) - 1} + \frac{n}{\log n} \\ T(n) = 2(2(2T(\frac{n}{8}) + \frac{\frac{n}{4}}{\log \frac{n}{4}}) + \frac{\frac{n}{2}}{\log \frac{n}{2}}) + \frac{n}{\log n} = 2^3T(\frac{n}{2^3}) + \frac{n}{\log(n) - 2} + \frac{n}{\log(n) - 1} + \frac{n}{\log n} \\ ... \\ T(n) = 2^kT(\frac{n}{2^k}) + \sum_{i = 0}^{ k - 1}\frac{n}{\log(n) - i} \\ \text{When } \frac{n}{2^k} = 1 \Rightarrow n = 2^k \Rightarrow k = \log_2(n) \\ T(n) = 2^{\log_2(n)}T(1) + \sum_{i = 0}^{ \log_2(n) - 1}\frac{n}{\log(n) - i} \\ T(n) = \Theta (n) + \sum_{i = 0}^{ \log_2(n) - 1}\frac{n}{\log_2(n) - i} \\ T(n) \approx \Theta (n) + \sum_{j = 1}^{ \log_2(n)}\frac{n}{j} \\ T(n) \approx \Theta (n) + nH_{\log_2(n)} \\ T(n) \approx \Theta (n) + n \ln(\log_2(n)) \\ \text{And finally } T(n)\ \in \Theta(n \ln(\log_2(n))) $$