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 2.3, Problem 3E
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To show that the solution of the recurrence relation is
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Solve 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) = O (n log n)
Consider the following recurrence relation: T(0) = 1, T(1) = 5, T(n) = 4T(n − 1) + 5T(n − 2) for n ≥ 2 Use the guess and check method to guess a closed form for T(n) and then prove that it is a closed form for T(n) using induction.
(a) For each of the following recurrences, give an expression for theruntime T (n) if the recurrence can be solved with the Master Theorem.Otherwise, indicate that the Master Theorem does not apply.(i) T (n) = T (n/2) + T(n/2) + T(n/2) + n2(ii) T (n) = 0.5T (n/2)+ 1/n(iii) T (n) = 3T (n/3) + n(iv) T (n) = 4T (n/2) + nlogn
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- Solve the recurrence relation using the iteration method: T(n)=4T(n/2) + (n^2)log narrow_forwardExpress the solution in big-O terms for the following recurrence relation: T(n) = 9*T(n/3) + n^3; T(1)=1; Answer: T(n) = O(......) .arrow_forwardSuppose that f(n) satisfies the divide-and-conquer recurrence relation f(n) = 3f(n/4)+n2/8 with f(1) = 2. What is f(64)?arrow_forward
- Solve 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_forwardSolve recurrence relation using substitution(i.e. proof by induction), to show that for some c > 0, T(n) ≤ cn2 for large enough n, where T(n) = 3T(n/2) + 2T(n)arrow_forwardSolve the following recurrences assuming that T(n) = Θ(1) for n ≤ 1. a) T (n) = 3T (n/π) + n/π b) T(n) = T(log n) + log narrow_forward
- Consider the recurrenceT (n) = 2T (n/2) + n log n.Provide a detailed proof why Master Theorem does not apply for this recurrence.arrow_forwardUse the master method to give tight asymptotic bounds for the following recurrence T(n) = 2T(n/4) + 1 Group of answer choices 1. ϴ(n0.5lgn) 2. ϴ(n0.5) 3. ϴ(n2) 4. ϴ(n)arrow_forwardExpand 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_forward
- Explain Master Theorem .Using Master Theorem solved the following recurrence. T(n) =4T(n/4) + log2 n b) T(n) = 4T(n/2) + n2arrow_forwardFind the order of growth for solutions of the following recurrences using master theorem. 1. T(n) = 4T(n/2) + n, T(1) = 1 2. T(n) = 4T(n/2) + n^2, T(1) = 1 3. T(n) = 4T(n/2) + n^3, T(1) = 1arrow_forwardGive tight asymptotic upper bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for sufficiently small n. Please give me a step by step answers, thanks.arrow_forward
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