Solution

The correct answer is \(\theta (n \log n)\).

Key Points

• The recurrence relation \(T(n) = 3T(n / 4) + n \log n\) can be solved using the Master Theorem.
• In the recurrence relation, \(a = 3\), \(b = 4\), and \(f(n) = n \log n\).
• Master Theorem states that for \(T(n) = aT(n / b) + f(n)\), the solution depends on the comparison between \(\log_b a\) and the growth rate of \(f(n)\).
• Here, \(\log_b a = \log_4 3\), and \(f(n) = n \log n\) has a growth rate of \(n^{1 + \epsilon}\), where \(\epsilon = 0\).
• Since \(f(n)\) dominates \(\log_4 3\), the solution is \(\theta (n \log n)\).

Step-by-Step Solution / Explanation

• Step 1: Identify \(a\), \(b\), and \(f(n)\) in the recurrence relation \(T(n) = 3T(n / 4) + n \log n\).
  • Here, \(a = 3\), \(b = 4\), and \(f(n) = n \log n\).
• Step 2: Compute \(\log_b a\), where \(b = 4\) and \(a = 3\).
  • \(\log_4 3 = \frac{\log 3}{\log 4}\).
  • Using approximation, \(\log 3 \approx 0.477\) and \(\log 4 \approx 0.602\).
  • \(\log_4 3 = \frac{0.477}{0.602} \approx 0.793.\)
• Step 3: Compare \(\log_b a\) and the growth rate of \(f(n)\).
  • \(f(n) = n \log n\) has a growth rate of \(n^{1 + \epsilon}\), where \(\epsilon = 0\).
  • \(\log_4 3 \approx 0.793\), which is less than \(1\), indicating that \(f(n)\) dominates.
• Step 4: Apply Master Theorem.
  • Since \(f(n)\) dominates \(\log_b a\), the solution to the recurrence relation is determined by \(f(n)\).
  • Thus, \(T(n) = \theta (n \log n)\).

Additional Information

• Master Theorem Conditions:
  • The recurrence relation must be of the form \(T(n) = aT(n / b) + f(n)\).
  • \(a\) and \(b\) must be constants, and \(f(n)\) should be asymptotically positive.
• Cases of Master Theorem:
  • If \(\log_b a < \text{growth rate of } f(n)\), solution depends on \(f(n)\).
  • If \(\log_b a = \text{growth rate of } f(n)\), solution includes logarithmic factors.
  • If \(\log_b a > \text{growth rate of } f(n)\), solution depends on \(\log_b a\).
• Applications:
  • Master Theorem is widely used to solve divide-and-conquer algorithms such as quicksort, mergesort, and binary search.