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 8E
Program Plan Intro
To find the asymptotically tight solution to the recurrence
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Use a recursion tree to determine a good asymptotic upper bound on the recurrence T(n)=4T(n/2+2)+n. Use the substitution method to verify your answer.
Use the recursion tree method to determine the asymptotic upper bound of T(n).
T(n) satisfies the recurrence T(n) = 2T(n-1) + c, where c is a positive constant, and
T(0)=0.
for 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.
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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- 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_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_forwardUsing 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_forward
- 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.arrow_forwardBuild a recursion tree for the following recurrence equation and then solve for T(n)arrow_forwardUse 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.arrow_forward
- Consider 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_forwardFor the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorthim: Algorithm Z divides an instance of size n into 2 subproblems, one with size n/4 and one with size n/5,recursively solves each one, and then takes O(n) time to combine the solutions and output the answer.arrow_forwardFor each of the following recurrences, give an expression for the runtime T(n) if the recurrence can be solved with the Master Theorem. Otherwise, indicate that the Master Theorem does not apply. For all cases, we have T(x) = 1 when x ≤ 100 (base of recursion). Ex.) T(n) = 3T(n/3) + √n We have nlog, a T(n) = O(n). a) T(n) = 5T(n/3) +2023n¹.6 b) T(n) = 9T(n/3) + 1984n² = n. Since f(n) = O(n¹-) (for any € < 1/2), we are at case 1 and n³ c) T(n) = 8T(n/2) + log n 4 d) T(n) = 16T(n/2) + n² log³ narrow_forward
- 2. Find the solution of the following recurrence equation by repeated substitution method, assuming n = 2¹ for some integer i. 1 T(n) = {4T (n/2) + n il if n = 1 ifn ≥ 2 3. Use the recursion tree to find the solution of the following recurrence. T(n)= 4T(n/4) + nªarrow_forwardAnswer 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_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
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