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 4.3, Problem 7E
Program Plan Intro
To show that a substitution proof with assumption
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We have already had a recurrence relation of an algorithm, which is T (n) = 4T (n/2) + n log n. We know T (1) ≤c.(a) express it as T (n) = O(f (n)), by using the iteration method.(b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T (n) at the iteration step t, and prove 3 that this form is correct for any iteration step t by using mathematical induction. Then by finding out the eventual number of t and substituting it into your general form of T (n), you get the O(·) notation of T (n).]
Expand the following recurrence to help you find a closed-form solution, and then use induction to
prove your answer is correct.
T(n) = √nT(√n) + n, for n>2; T(2) = 1.
Practice 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() -n²logn
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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- Solve 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_forwardFor 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_forwardSolve the recurrence below in the same style as done in lecture. Simplify any formula you get. T(1) = 4 T(n) = n - 3 + T(n-1) for any n > 1.arrow_forward
- Use the substitution method to prove that the re- currence T (n) = T (n − 1) + Θ(n) has the solution T(n) = Θ(n2) as claimed in classarrow_forwardUse 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)arrow_forwardSolve 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) = 0 (n log n). Solve the recurrence relation T (n) = T ( √n) + c. n > 4 Derive the runtime of the below codearrow_forward
- Solving recurrences using the Master method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the Master method. Assume that T(n) is constant for n<=3. Make your bounds as tight as possible and justify your answers. Solving recurrences using the Master method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the Master method. Assume that T(n) is constant for n<=3. Make your bounds as tight as possible and justify your answers.f. T(n)=T(\sqrt()n)+\Theta (lglgn)g. T(n)=10T((n)/(3))+17n^(1.2)h. T(n)=7T((n)/(2))+n^(3)i. T(n)=49T((n)/(25))+(\sqrt()n)^(3)lgnj. T(n)=4T((n)/(2))+lognarrow_forwardLet the statement be "If n is not an odd integer then square of n is not odd.", then if P(n) is "n is an not an odd integer" and Q(n) is "(square of n) is not odd." For contrapositive proof, we should prove. O vn not (P(n) → Q(n)) O a (P (n) – Q(n) ) Ovn (P(n) – Q(n) ) O vn ( not Q(n) → not P(n) )arrow_forwardSolve 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_forward
- 1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.arrow_forwardSuppose f(n) = O(F(n)) and g(n) = O(G(n)). Is it true that f(n) / g(n) = O(F(n) / G(n))? Provide a proof using the definition of big-oh, or provide values for the four functions as a counterexample.arrow_forward4. Practice with the iteration method. We have already had a recurrence relation ofan algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c.(a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.Answer:(b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction.Then by finding out the eventual number of t and substituting it into your generalform of T(n), you get the O(·) notation of T(n).]arrow_forward
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