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 6E
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To prove that the solution to the recurrence
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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.
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).
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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- 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.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_forwardUse a recursion tree to determine a good asymptotic upper bound on following recurrences. Please see Appendix of your text book for using harmonic and geometric series. a) T (n) = T(n/5) + O(n)2 b) T (n) = 10T(n/2) + O(n)2 c) T (n) = 10T(n/2) + Θ (1) d) T (n) = 2T (n/2) + n/ lg n e) T (n) = 2T (n - 1) + Θ (1)arrow_forward
- Solve each of the following recurrences using the Iterative Method, Master Theorem, and Recursion Tree. 1. T(n) = T()+ 1, T(1) = 1 (you may assume that n = 2") 2. T(n) = T(n - 1) + log, n, T(0) =1 |arrow_forwardGiven 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 Xarrow_forwardProblem 3. Use recursion tree to solve the following recurrence. T(n) = T(n/15) +T(n/10) +2T(n/6) + /narrow_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 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_forwardLet S = {3n :neZ}.A recursive definition for the set S is: %3D Basis Step: 3 ES Recursive Step: If x E S then a +3 E S Prove by structural induction that for every x E S, x+x E S. Hint: Use the recursive definition of S to set up your proof by structural induction and use the definition S = {3n : n e Z*} in your proof. + Drag and drop an image or PDF file or click to browse...arrow_forward
- 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_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_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) andarrow_forward
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