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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Give the solution for T(n) in the following recurrence. Assume that T(n) is constant for small n. Provide brief justification for the answer.
Use the substitution method to show that the recurrence defined by T(n) = 2T(n/3) + Θ(n) hassolution T(n) = Θ(n).
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)
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Introduction to Algorithms
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- 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 )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_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_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_forward1. Use the substitution method to show the recurrence: T(n) = 4T(n/2) + (n) has solution T(n) = O(n²)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_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_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_forwardSolve the recurrence: T(n) = T(n/2) + 4n T(1) = 1arrow_forward
- Use the substitution method to find the solution of following recurrences.T(n) = T( n / 2) + Carrow_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_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_forward
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