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 4.3, Problem 8E
Program Plan Intro
To show that a substitution proof with assumption
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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) 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 3 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).]
Expand the following recurrence to help you find a closed-form solution, and then use induction to
prove your answer is correct.
T(n) = √nT(√n) + n, for n>2; T(2) = 1.
Expand the following recurrence to help you find a closed-form solution, and then use induction to prove your answer is correct.
T(n) = T(n−1) + 5 for n > 0; T(0) = 8.
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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- Practice Exercise #3: For 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. 1. T(n) = T + 2⁰ 2. T(n) = √2T) + logn 3T (+2 3. T(n) = 4. T(n) = 64T() -n²lognarrow_forwardSolve the following recurrence equations by expanding the formulas (also called the 'iteration method' on slides). Specifically, you should get T(n) = O(f(n)) for a function f(n). You may assume that T(n) = O(1) for n = O(1). You should not use the Master Theorem. (a) T(n) = 2T (n/3) + 1. (b) T(n) = 7T(n/7) + n. (c) T(n) = T(n − 1) + 2.arrow_forwardGive asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T (n) is constant for n ≤ 3. Make your bounds as tight as possible, and justify your answers (you can use any of the methods we discussed in class). 1). T(n)=T(n/2)+lgn.arrow_forward
- For 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_forwardSolving recurrences using the Master method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the Master method. Assume that T(n) is constant for n<=3. Make your bounds as tight as possible and justify your answers. Solving recurrences using the Master method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the Master method. Assume that T(n) is constant for n<=3. Make your bounds as tight as possible and justify your answers.f. T(n)=T(\sqrt()n)+\Theta (lglgn)g. T(n)=10T((n)/(3))+17n^(1.2)h. T(n)=7T((n)/(2))+n^(3)i. T(n)=49T((n)/(25))+(\sqrt()n)^(3)lgnj. T(n)=4T((n)/(2))+lognarrow_forwardUse the master method to give tight asymptotic bounds for the following recurrence T(n) = 2T(n/4) + nº.5 (nº.5Ign) e(nº.5) e(n) ○ e(n²)arrow_forward
- 4. Practice with the iteration method. We have already had a recurrence relation ofan 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 generalform of T(n), you get the O(·) notation of T(n).]arrow_forwardCan the master method be applied to the recurrence T(n) = 4T(n²) + n²logn Why or why not? Give an asymptotic upper bound for this recurrence.arrow_forwardSolve the recurrence relation: T (n) = T (n/2) + T (n/4) + T (n/8) + n. Use the substitution method, guess that the solution is T (n) = 0 (n log n). Solve the recurrence relation T (n) = T ( √n) + c. n > 4 Derive the runtime of the below codearrow_forward
- Find a tight bound solution for the following recurrence: T(n) = 4 T(n/2) + c n (c is a positive constant) That is, find a function g(n) such that T(n) ∊ Θ(g(n)). For convenience, you may assume that n is a power of 2, i.e., n=2k for some positive integer k. Justify your answer. [Note: Read question 1-(2) first before writing your answer] Prove your answer in 1-(1) either using the iteration method or using the substitution method.arrow_forwardSolve the recurrence below in the same style as done in lecture. Simplify any formula you get. T(1) = 4 T(n) = n - 3 + T(n-1) for any n > 1.arrow_forwardSolve the following recurrences exactly:(a) T(1) = 8, and for all n ≥ 2, T(n) = 3T(n − 1) + 15.(b) T(1) = 1, and for all n ≥ 2, T(n) = 2T(n/2) + 6n − 1 (n is a power of 2)arrow_forward
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