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 7.2, Problem 1E
Program Plan Intro
To prove that the solution of the recurrence relation
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Solve the recurrence relation: T (n) = T (n/2) + T (n/4) + T (n/8) + n. Use the substitution
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Use 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)
Use the substitution method to show that for the recurrence equation:
T( 1 )=1
T( n )=T( n/3 ) + n the solution is T( n )=O ( n )
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Introduction to Algorithms
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- Use the substitution method to show that the recurrence defined by T(n) = 2T(n/3) + Θ(n) hassolution T(n) = Θ(n).arrow_forwardSolve the first-order linear recurrence T(n) = 3T(n − 1) +8, T(0) = 6 by finding an explicit closed formula for T(n) and enter your answer in the box below. T(n) =arrow_forward4. Consider the recurrence: T(n) = T(n/2) + T(n/4) + n, T(m) = 1 for m <= 5. Use the substitution method to give a tight upper bound on the solution to the recurrence using O-notation.arrow_forward
- Please solve using iterative method: Solve the following recurrences and compute the asymptotic upper bounds. Assume that T(n) is a constant for sufficiently small n. Make your bounds as tight as possible. a. T(n) = T(n − 2) + √n b.T(n) = 2T(n − 1) + carrow_forwardUse the substitution method to obtain the exact value of T(n) in the following recurrence. Show all computations. T(1) = 1, T(n)=T(n-1)+3n,n>1arrow_forward1. Use the substitution method to show the recurrence: T(n) = 4T(n/2) + (n) has solution T(n) = O(n²)arrow_forward
- 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.arrow_forwardSolve the recurrence: T(n) = T(n/2) + 4n T(1) = 1arrow_forward7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).arrow_forward
- Use the substitution method to find the solution of following recurrences.T(n) = T( n / 2) + Carrow_forwardPractice 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_forward
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