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.
(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).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb3095d4-472d-42c8-847a-ae9732c32b9b%2F083ed480-f0dc-4156-8f29-e72aa489eeec%2Fjweuy6r_processed.png&w=3840&q=75)
Iteration method :
A "brute force" approach to solving a recurrence relation is the iteration technique. The fundamental concept is to repeatedly substitute the recurrent component's value until a pattern (often a summation) emerges, at which time the summation may be used to assess the recurrence.
in this Expanding the recurrence and expressing it as a sum of the terms of n and the beginning condition is what it implies.
Step by step
Solved in 3 steps with 2 images
