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
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To prove that the solution of the recurrence relation
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Question 3
Please solve the recurrence and show its proof by induction
of:
T(1) = 3
T(n) = T(n/3) + 2n, n > 1
(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
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)
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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_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_forwardWhat is the complexity of the following recurrence?arrow_forward
- for the given 1,2,3 find the recurrences - the closed-form expression for n. 1) S(0) = 6 for n = 0 S(n) = S(n-1) + 2 for n = 1, 2, 3, ... 2) T(1) = 2 for n = 1 T(n) = 2T(n-1) + 4 for n = 2, 3, 4, … 3) Q(1) = c for n = 1 Q(n) = Q(n/2) + 2n for n = 2, 4, 8, …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_forwardUse the substitution method to show that the recurrence defined by T(n) = 2T(n/3) + Θ(n) hassolution T(n) = Θ(n).arrow_forward
- How many lines will the following function write? Write the recurrence relation of each and solve using the Master theorem. Give your answer as a function of n (in the form Θ( · )).arrow_forwardConsider 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.arrow_forwardFind the solution for each of the following recurrences, and then give tight bounds (i.e., in Θ(·)) for T (n). (a) T (n) = T (n − 1) + 1/n with T (0) = 0. (b) T (n) = T (n − 1) + cn with T (0) = 1, where c > 1 is some constant (c) T (n) = 2 T (n − 1) + 1 with T (0) = 1arrow_forward
- Solve 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_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_forwardI am not an engineering student. Grateful for your detailed explanation.Give tight asymptotic bounds for the following recurrences. • T(n) = 4T(n/2) + n 3 − 1. • T(n) = 8T(n/2) + n 2 . • T(n) = 6T(n/3) + n. • T(n) = T( √ n) + 1arrow_forward
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