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 4.4, Problem 9E
Program Plan Intro
To find the asymptotically tight solution to the recurrence
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Problem 3.
Consider the following recurrence.
T(n) = {(n) = 37(n
T(n) = 3T(n/2) + n²
if n=1
otherwise.
(a) Solve this recurrence exactly by the method of substitution. You may assume n is a
power of 2.
(b) Solve it using the recursion tree method.
for the following problem we need to use a recursion tree. so we can determine an asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. the substitution method must be used to solve.
Consider the recurrence T(n).
r(n) = { T[{\√~]) + d
if n ≤ 4
([√n])+d_ifn>4
Use the recursion tree technique or repeated substitution to come up with a good
guess of a tight bound on this recurrence and then prove your tight bound correct with
induction or another technique.
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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- For each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 2T(N-1) + 1 T(N) = 3T(N-1) + narrow_forwardSolving recurrences using the Substitution method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the substitution method. Assume that T(n) is constant n ≤ 2. Make your bounds as tight as possible and justify your answers. Hint: You may use the recursion trees or Master method to make an initial guess and prove it through induction a. T(n) = 2T(n-1) + 1 b. T(n) = 8T(n/2) + n^3arrow_forwardFor the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorthim: Algorithm Z divides an instance of size n into 2 subproblems, one with size n/4 and one with size n/5,recursively solves each one, and then takes O(n) time to combine the solutions and output the answer.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_forwardRecurrence relations: Master theorem for decreasing functions T(n) = {₁T(n- aT(n −b) + f(n), if n = 0 if n > 0 f(n) = nd What is T(n)?arrow_forwardWrite a recurrence relation describing the WORST CASE running time of each of the following algorithms. Justify your solution with either substitution, a recursion tree, or induction. Simplify your answer to O(nk) or O(nk log(n)) whenever possible. If the algorithm takes exponential time, then just give exponential lower bounds. (b) 1: function Func(A[],n) 2: if n < 10 then return A|1| 4: end if 5: X = 6: for i = 1 to n3/2 do 7: x = x + A[[i/n]] 8: end for 9: x = x + Func(A[],n – 5) 10: return x 11: end function 3:arrow_forward
- Solve each of the following recurrences using the Iterative Method, Master Theorem, and Recursion Tree. 1. T(n) = T()+ 1, T(1) = 1 (you may assume that n = 2") 2. T(n) = T(n - 1) + log, n, T(0) =1 |arrow_forward(b) Solve the recurrence relation : T (n) =T O(n). 3 using the recursion tree method.arrow_forwardLet S = {3n :neZ}.A recursive definition for the set S is: %3D Basis Step: 3 ES Recursive Step: If x E S then a +3 E S Prove by structural induction that for every x E S, x+x E S. Hint: Use the recursive definition of S to set up your proof by structural induction and use the definition S = {3n : n e Z*} in your proof. + Drag and drop an image or PDF file or click to browse...arrow_forward
- Please explain Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) and (b) using Master Theoremarrow_forwardFor the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorrthim Problem: Algorithm V divides an instance of size n into 6 subproblems of size n/6 each, recursively solves each one, and then takes O(n) time to combine the solutions and output the answer.arrow_forward(c) Draw the recursion tree for the recurrence T(n) = 3T (|n/V3|) + cn, where c is a constant, and determine a good asymptotic upper bound on its solution. Verify your bound by the substitution method.arrow_forward
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