Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method

Question

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n)=4T\left(\frac{n}{2}\right)+n^2$$ My guess is $T(n)$ is $\Theta (n\log n)$ (and I am sure about it because of master theorem), and to find an upper bound, I use induction. I tried to show that $T(n)\leq cn^{2}\log n$ but that did not work, I got $T(n)\leq cn^{2}\log n+n^{2}$. Then I tried to show that, if $T(n)\leq c_{1}n^{2}\log n-c_{2}n^{2}$, then it is also $O(n^{2}\log n)$, but that also did not work and I got $T(n)\leq c_{1}n^{2}\log(n/2)-c_{2}n^{2}+n^{2}$. What trick can I do to show that? Thanks

Answer 1

If $T(\frac{n}{2}) \leq c(\frac{n}{2})^2\log_2(\frac{n}{2})+T(1)$, then \begin{align} T(n)=4T\left(\frac{n}{2}\right)+n^2 & \leq 4c\left(\frac{n}{2}\right)^2\log_2\left(\frac{n}{2}\right)+4T(1)+n^2 \\ &=cn^2\log_2(n)-cn^2+4T(1)+n^2 \\ &\leq cn^2\log_2(n)+T(1) \end{align} for $c \geq 1+3T(1)$.

Answer 2

By way of enrichment we solve another closely related recurrence that admits an exact solution. Suppose we have $T(0)=0$ and for $n\ge 1$ (this gives $T(1)=1$) $$T(n) = 4 T(\lfloor n/2 \rfloor) + n^2.$$

Furthermore let the base two representation of $n$ be $$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k.$$

Then we can unroll the recurrence to obtain the following exact formula for $n\ge 1$ $$T(n) = \sum_{j=0}^{\lfloor \log_2 n \rfloor} 4^j \left( \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^{k-j} \right)^2 = \sum_{j=0}^{\lfloor \log_2 n \rfloor} \left(\sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^k\right)^2.$$

Now to get an upper bound consider a string of one digits which yields $$T(n) \le \sum_{j=0}^{\lfloor \log_2 n \rfloor} \left(\sum_{k=j}^{\lfloor \log_2 n \rfloor} 2^k\right)^2.$$

Note that this bound is attained and cannot be improved. It simplifies to $$\left(4 \lfloor \log_2 n \rfloor - \frac{8}{3}\right) \times 4^{\lfloor \log_2 n \rfloor} + 4 \times 2^{\lfloor \log_2 n \rfloor} - \frac{1}{3}.$$

The lower bound is for the case of a one digit followed by a string of zeros and yields $$T(n) \ge \sum_{j=0}^{\lfloor \log_2 n \rfloor} 2^{2\lfloor \log_2 n \rfloor}.$$ It simplifies to $$(1+\lfloor \log_2 n \rfloor) 2^{2\lfloor \log_2 n \rfloor}.$$

Joining the dominant terms of the upper and the lower bound we obtain the asymptotics $$\color{#006}{\lfloor \log_2 n \rfloor 2^{2 \lfloor \log_2 n \rfloor} \in \Theta\left(\log_2 n \times 2^{2 \log_2 n}\right) = \Theta\left(\log n \times n^2\right)}.$$

These are both in agreement with what the Master theorem would produce.

This MSE link has a series of similar calculations.