The running time of an algorithm T, where 'n' is the input size, is given by- T(1)= 1, if n=1 =2T([n/2])+vn,for n≥2

Find out the order of this algorithm.

To determine the order (or asymptotic complexity) of the algorithm described by the recurrence relation:

\[
T(1) = 1, \quad \text{if } n = 1
\]
\[
T(n) = 2T\left(\frac{n}{2}\right) + vn, \quad \text{for } n \geq 2
\]

where \(v\) is a constant.

### Step 1: Analyze the Recurrence Relation
The recurrence can be solved using the **Master Theorem**. The recurrence is of the form:

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

where:
- \(a = 2\),
- \(b = 2\),
- \(f(n) = vn\).

### Step 2: Apply the Master Theorem
The Master Theorem gives us the order of the recurrence based on the comparison between \(f(n)\) and \(n^{\log_b a}\).

Here, \(\log_b a = \log_2 2 = 1\).

- Case 1: If \(f(n) = O(n^c)\) where \(c < \log_b a\), then \(T(n) = \Theta(n^{\log_b a})\).
- Case 2: If \(f(n) = \Theta(n^{\log_b a} \cdot \log^k n)\) where \(k \geq 0\), then \(T(n) = \Theta(n^{\log_b a} \cdot \log^{k+1} n)\).
- Case 3: If \(f(n) = \Omega(n^c)\) where \(c > \log_b a\) and if \(a f\left(\frac{n}{b}\right) \leq k f(n)\) for some constant \(k < 1\), then \(T(n) = \Theta(f(n))\).

### Step 3: Determine the Case
Here, \(f(n) = vn\) and \(\log_b a = 1\), so \(f(n) = \Theta(n^1)\).

This fits **Case 2** of the Master Theorem, where \(c = \log_b a\) and \(f(n) = \Theta(n \log^0 n)\).

Therefore, according to Case 2, the running time \(T(n)\) is given by:

\[
T(n) = \Theta(n \log n)
\]

### Conclusion
The order of the algorithm is \( \Theta(n \log n) \).

---

Do you want more details on how the Master Theorem was applied or have any other questions?

Here are some related questions:
1. What is the Master Theorem and how is it used in solving recurrences?
2. Can you explain the different cases of the Master Theorem with examples?
3. How would the order change if the recurrence relation had a different form for \(f(n)\)?
4. What if \(T(n)\) was divided into more than two subproblems?
5. How does the constant \(v\) affect the final time complexity?

**Tip:** The Master Theorem is a powerful tool for solving divide-and-conquer recurrences, but it only applies when the recurrence fits the required form. Always check the conditions carefully before applying it.

Explore how to determine the order of an algorithm with a detailed explanation of the Master Theorem. Solve the recurrence relation T(1) = 1, T(n) = 2T(n/2) + vn to find the asymptotic complexity. This analysis covers key concepts in algorithm analysis and is suitable for advanced college-level students.

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