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4. Practice with the iteration method. We have already had a recurrence relation of an algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c. (a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.

4. Practice with the iteration method. We have already had a recurrence relation of an algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c. (a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.

C++ Programming: From Problem Analysis to Program Design
C++ Programming: From Problem Analysis to Program Design
8th Edition
ISBN:9781337102087
Author:D. S. Malik
Publisher:D. S. Malik
Chapter15: Recursion
Section: Chapter Questions
Problem 18SA
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(b) Prove, by using mathematical induction, that the iteration rule you have
observed in 4(a) is correct and you have solved the recurrence relation correctly.
[Hint: You can write out the general form of T(n) at the iteration step t, and prove
that this form is correct for any iteration step t by using mathematical induction.
Then by finding out the eventual number of t and substituting it into your general
form of T(n), you get the O(·) notation of T(n).

See image for reference to part a, the answer to part a came out to be O(n^2). Need help with b?

4. Practice with the iteration method. We have already had a recurrence relation of an algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c. (a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method. Answer: (b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction. Then by finding out the eventual number of t and substituting it into your general form of T(n), you get the O(-) notation of T(n).]
Transcribed Image Text:4. Practice with the iteration method. We have already had a recurrence relation of an algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c. (a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method. Answer: (b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction. Then by finding out the eventual number of t and substituting it into your general form of T(n), you get the O(-) notation of T(n).]
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