To analyze the recurrence relation $T(n) = 2T\left(\frac{n}{2}\right) + n \log n$ using the Master Theorem, let's break down the parameters and determine if the theorem applies.

Given:

• $T(n) = 2T\left(\frac{n}{2}\right) + n \log n$
• Parameters:
  • $a = 2$
  • $b = 2$
  • $f(n) = n \log n$

Applying the Master Theorem

The Master Theorem applies to recurrence relations of the form: $T(n) = a T\left(\frac{n}{b}\right) + f(n)$ where $a \geq 1$, $b > 1$, and $f(n)$ is a function of $n$.

We compute:

1. Calculate $n^{\log_b a}$: $\log_b a = \log_2 2 = 1$ So, $n^{\log_b a} = n$.

2. Compare $f(n) = n \log n$ with $n^{\log_b a} = n$:

   • We check three cases:
     1. Case 1 (if $f(n) = O(n^{\log_b a - \epsilon})$ for some $\epsilon > 0$): This does not apply since $n \log n$ grows faster than $n^{1 - \epsilon}$ for any $\epsilon > 0$.
     2. Case 2 (if $f(n) = \Theta(n^{\log_b a})$): This does not apply either since $f(n) = n \log n$ is not $\Theta(n)$ but slightly larger.
     3. Case 3 (if $f(n) = \Omega(n^{\log_b a + \epsilon})$ for some $\epsilon > 0$ and $a f\left(\frac{n}{b}\right) \leq c f(n)$ for some $c < 1$): This does not apply because $n \log n$ is not polynomially larger than $n$.

Since none of these cases fit perfectly, the Basic Form of the Master Theorem does not apply directly to this recurrence relation.

Solution Approach Outside the Master Theorem

For recurrence relations like $T(n) = 2T\left(\frac{n}{2}\right) + n \log n$, where $f(n)$ grows slightly faster than $n^{\log_b a}$ but does not satisfy Case 3, the Regularity Condition fails, and alternative methods (such as the recursion-tree method or substitution method) are usually applied. The solution is often shown to be $T(n) = \Theta(n \log^2 n)$ for such cases.

Would you like a detailed breakdown of this alternative method?

Follow-up Questions:

1. What other types of recurrence relations are solved using the Master Theorem?
2. How does the recursion-tree method help in solving recurrence relations?
3. What are the limitations of the Master Theorem in divide-and-conquer algorithms?
4. Can $n \log n$ functions fit into any advanced versions of the Master Theorem?
5. How would substitution work for verifying $T(n) = \Theta(n \log^2 n)$?

Tip:

For recurrence relations with $n \log n$ terms, remember that functions slightly larger than polynomial growth may need alternative approaches beyond the Master Theorem.