T(n) = 2T(n/2) + n. solve this recurrence relation with any method

To solve the recurrence relation \( T(n) = 2T(n/2) + n \), we can use the **Master Theorem**, which provides a way to determine the time complexity of recurrence relations of this form.

The given recurrence relation is:

\[ T(n) = 2T(n/2) + n \]

**Step 1: Identify the components of the Master Theorem.**

The general form of the Master Theorem is:

\[ T(n) = aT\left(\frac{n}{b}\right) + f(n) \]

Where:
- \( a \) is the number of subproblems at each level (in this case, \( a = 2 \)).
- \( b \) is the factor by which the problem size is reduced at each step (in this case, \( b = 2 \)).
- \( f(n) \) is the cost of the work done outside the recursive calls (in this case, \( f(n) = n \)).

**Step 2: Calculate \( n^{\log_b a} \).**

We need to find \( n^{\log_b a} \). In this case:

\[ a = 2, \quad b = 2 \]

\[ \log_b a = \log_2 2 = 1 \]

So:

\[ n^{\log_b a} = n^1 = n \]

**Step 3: Compare \( f(n) \) with \( n^{\log_b a} \).**

Here, \( f(n) = n \) and \( n^{\log_b a} = n \).

Since \( f(n) = \Theta(n^{\log_b a}) \), we can use Case 2 of the Master Theorem, which states:

- If \( f(n) = \Theta(n^{\log_b a} \log^k n) \) for some \( k \geq 0 \), then \( T(n) = \Theta(n^{\log_b a} \log^{k+1} n) \).

In this problem, \( k = 0 \), so:

\[ T(n) = \Theta(n^{\log_b a} \log n) = \Theta(n \log n) \]

**Conclusion:**

The time complexity of the given recurrence relation is \( \Theta(n \log n) \).

Would you like more details or have any questions?

### Follow-up Questions
1. What is the significance of the Master Theorem in analyzing algorithms?
2. Can you explain Case 1 and Case 3 of the Master Theorem?
3. How would the time complexity change if \( f(n) \) were \( \Theta(n^2) \)?
4. What is the meaning of \( \Theta \) notation?
5. How do we derive the logarithmic factor in \( n \log n \)?
6. Can you provide an example of a problem that has a similar time complexity?
7. What is the difference between \( \Theta \), \( O \), and \( \Omega \) notations?
8. How does the divide-and-conquer method relate to the Master Theorem?

### Tip
When analyzing algorithms, always pay attention to the recursive part and the non-recursive work done at each level; this helps determine the overall complexity efficiently.

Learn how to solve the recurrence relation T(n) = 2T(n/2) + n using the Master Theorem. Understand the complexity analysis of recursive algorithms with detailed step-by-step explanations. Suitable for advanced students and professionals.

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