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
Program Plan Intro
To prove that the solution to the recurrence
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For 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) = 4T(N/2) + n2logn
T(N) = 5T(N/2) + n2/logn
Use 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)
For 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) + n
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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- Please 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_forwardUse recursion tree to guess a bound, then proof it using induction. Finally, use master theorem (if applicable) to directly get the bound. Try to make your bounds as tight as possible. T(n) = 2T(n/2) + n T(n) = 2T(n-2) + narrow_forwardConsider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forward
- Using the substitution method, Prove that the running time of the following recursive relation is O(N) T(n) = T(n / 2) + T(7n / 10) + O(n)arrow_forwardFor each, draw the recursion tree, find the height of the tree, the running time of each layer, and the sum of running times. Then use this info to find the explicit answer for T(n). a. T(n) = 2T(n/4) + √ n (n is a power of 4 (n = 4^k) for some positive integer k) b. T(n) = 9T(n/3) + n^2 (n is a power of 3 (n = 3^k) for some positive integer k) c. T(n) = T(n/2) + 1 (n is a power of 2 (n = 2^k) for some positive integer k)arrow_forwardSolve the following recurrences using recursion tree method and write the asymptotic time-complexity. 1. T(n) = 3T (n/4) + n^22. T(n) = T (n/5) + T(4n/5) + n3. T(n) = 3T(n − 1) + n^4 4. T(n) = T (n/2) + n^2arrow_forward
- Plz solve correctly and don't use chat gpt. Give a recursive definition for the set of all strings of 0’s and 1’s that have more 0’s than 1’s.arrow_forwardDraw a recursion tree for a recurrence and use the Substitution Method to prove the solution. ( make a sample question )arrow_forwardProvide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-Θ by determining a formula that counts the number of nodes in the tree as a function of n.arrow_forward
- The function f is defined for non-negative integers a and b recursively as follows:f(a, b) ={0 if a = 0 or b = 0f(a − 1, b − 1) + 2a − 1 if a = bf(a − b, b) + f(b, b) if a > bf(a, a) + f(b − a, a) if a < b}Compute f (3, 2) by drawing a recursion tree showing all of the computationrequired and then use your tree to compute the answer.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_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_forward
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