To solve the recurrence relation $T(n) = 3T(n/2) + n \log n$ with the base case $T(1) = 1$, we can use the Master Theorem, which is a powerful tool for analyzing divide-and-conquer algorithms.

Step 1: Identify the parameters

We can compare our recurrence to the standard form: $T(n) = aT(n/b) + f(n)$ where:

• $a = 3$
• $b = 2$
• $f(n) = n \log n$

Step 2: Calculate $n^{\log_b a}$

Next, we compute $n^{\log_b a}$:

• $\log_b a = \log_2 3$

Using the change of base formula: $\log_2 3 \approx 1.585$ Thus, $n^{\log_b a} = n^{\log_2 3} \approx n^{1.585}$

Step 3: Compare $f(n)$ and $n^{\log_b a}$

Now we need to compare $f(n) = n \log n$ with $n^{\log_2 3}$.

• $f(n) = n \log n$ grows slower than $n^{1.585}$ as $n \to \infty$. This is because $\log n$ grows slower than any polynomial function.

Step 4: Check the conditions of the Master Theorem

According to the Master Theorem, we are in Case 1 if $f(n)$ is polynomially smaller than $n^{\log_b a}$: $f(n) = O(n^{\log_b a - \epsilon}) \text{ for some } \epsilon > 0$ In this case: $n \log n = O(n^{1.585 - \epsilon}) \text{ for } \epsilon = 0.585$ This condition is satisfied.

Step 5: Apply Master Theorem

Since all conditions for Case 1 hold, we conclude that: $T(n) = \Theta(n^{\log_2 3}) = \Theta(n^{1.585})$

Final Result

Thus, the solution to the recurrence relation $T(n) = 3T(n/2) + n \log n$ is: $T(n) = \Theta(n^{1.585})$

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If you have any questions or would like further details, please let me know! Here are a few related questions for further exploration:

1. How does the value of $a$ and $b$ affect the solution of the recurrence?
2. Can you provide an example of a divide-and-conquer algorithm that follows a similar recurrence?
3. What are other methods to solve recurrences apart from the Master Theorem?
4. How does the growth rate of $f(n)$ influence the choice of which case in the Master Theorem applies?
5. What are practical applications of understanding such recurrence relations in algorithm analysis?

Tip: Always verify the growth rates of your functions when applying the Master Theorem to ensure you are applying the correct case.