Question

Substitution Method: Use the substitution method to show that T(n) is O(n log? (n)): • T(n) = 2T(n/2) + n. Solution: Recursion tree Method: Solve T(n) = T(n/4) + T(n/2) + n²: n² (n/4)² (n/2)² (n/16)² (n/8)² (n/8)² (n/4)² ... ?(1) Total = n²\left(1 + \frac{5}{16} + \left(\frac{5}{16}\right)² + \left(\frac{5}{16}\right)³ + ...\right) = ?(n²) geometric series

          Substitution Method:
Use the substitution method to show that T(n) is O(n log? (n)):
• T(n) = 2T(n/2) + n.
Solution:
Recursion tree Method:
Solve T(n) = T(n/4) + T(n/2) + n²:
n²
(n/4)²
(n/2)²
(n/16)² (n/8)² (n/8)² (n/4)² ...
?(1)
Total = n²\left(1 + \frac{5}{16} + \left(\frac{5}{16}\right)² + \left(\frac{5}{16}\right)³ + ...\right)
= ?(n²) geometric series

Substitution Method:
Use the substitution method to show that T(n) is O(n log? (n)):
• T(n) = 2T(n/2) + n.
Solution:
Recursion tree Method:
Solve T(n) = T(n/4) + T(n/2) + n²:
n²
(n/4)²
(n/2)²
(n/16)² (n/8)² (n/8)² (n/4)² ...
?(1)
Total = n²(1 + (5)/(16) + ((5)/(16))² + ((5)/(16))³ + ...)
= ?(n²) geometric series

Added by Robert Z.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Rosemary Charnley

Step 1

To solve the given recurrence relation \(T(n) = T(n/4) + T(n/2) + n^2\) using the recursion tree method, follow these steps: ### Show more…

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Substitution Method: Use the substitution method to show that T(n) is O(n log n). T(n) = 2T(n/2) + n. Solution: Recursion tree Method: Solve T(n) = T(n/4) + T(n/2) + n^2. 1 n^2 n/4 n/2^2 n/16^2 n/8^2 n/8 n/4 256 ... O(1) Total = n^1 + 16 = (n geometric series)