Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 4.4, Problem 3E
Program Plan Intro
To determine the good asymptotic upper bound of the recurrence relation
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Use a recursion tree to determine a good asymptotic upper bound on the recurrenceT(n) = 3T(n/3) + n.You can assume that n is a power of 3.Show all your work.
Given the recurrence relation:
• T(n) = 8 if n 6.
Find the value of T(495).
[Hint: Use a recursion tree to solve the recurrence exactly and plug the argument into the function you obtain.]
Answer: 493
The correct answer is: 336
X
Use a recursion tree to determine a good asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. Use the substitution method to prove your answer.
Chapter 4 Solutions
Introduction to Algorithms
Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Prob. 5E
Ch. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Prob. 9ECh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Prob. 3ECh. 4 - Prob. 1PCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Prob. 6P
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- Using a recursion tree, show the process how to solve the following recurrence in terms of the big O representation. Use the substitution method to verify your result. T(n) = T(n/2)+T(n/3)+cnarrow_forwardSolving recurrences using the Substitution method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the substitution method. Assume that T(n) is constant n ≤ 2. Make your bounds as tight as possible and justify your answers. Hint: You may use the recursion trees or Master method to make an initial guess and prove it through induction a. T(n) = 2T(n-1) + 1 b. T(n) = 8T(n/2) + n^3arrow_forwardProblem 3. Use recursion tree to solve the following recurrence. T(n) = T(n/15) +T(n/10) +2T(n/6) + /narrow_forward
- Answer the following for the recurrence T(n) = T( n / 2 ) + T( n / 4 ) + n. (a) Use the Recursion Tree method to guess the upper-bound. (b) Prove by induction the upper-bound obtained in the previous question (problem a).arrow_forwardProblem 3. Consider the following recurrence. T(n) = {(n) = 37(n T(n) = 3T(n/2) + n² if n=1 otherwise. (a) Solve this recurrence exactly by the method of substitution. You may assume n is a power of 2. (b) Solve it using the recursion tree method.arrow_forwardFor each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 2T(N-1) + 1 T(N) = 3T(N-1) + narrow_forward
- Use the recursion tree method to solve the following recurrence T(n) by finding the tightest function f(n) such that T(n) = O(f(n)). T(n) ≤ 4.T(n/3) +0(n³)arrow_forwardfor the following problem we need to use a recursion tree. so we can determine an asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. the substitution method must be used to solve.arrow_forwardPlease explain Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) and (b) using Master Theoremarrow_forward
- Using the recursion tree method, show to work to derive the runtime for the following recurrence relation: Hint:The resulting runtime should should be O(n*n^(1/log3(4/3))) or O(n^4.8188). need the process how to reach the answer and that was the whole question Note: will like the correct and detailed answer. Thank you!arrow_forwardQuestion 5: Using recursion tree method and then substitution method to solve T(n) = T(n/3) + T(n/4) Show your workarrow_forwardConsider the recurrence T(n). r(n) = { T[{\√~]) + d if n ≤ 4 ([√n])+d_ifn>4 Use the recursion tree technique or repeated substitution to come up with a good guess of a tight bound on this recurrence and then prove your tight bound correct with induction or another technique.arrow_forward
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