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 3.2, Problem 3E
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To prove the equation 3.19 and also to prove that
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Prove or disprove that if f(n) = (g(n)), then 4f(n) = (49(n)).
(b)
Prove: max(f(n), g(n)) E O(f(n)g(n)), i.e., s(n) E O(f(n)g(n)).
Assume for all natural n, f(n) > 1 and g(n) > 1. Let s(n) = max(f(n), g(n)).
Given f(n) ∈ Θ(n), prove that f(n) ∈ O(n²).
Given f(n) ∈ O(n) and g(n) ∈ O(n²), prove that f(n)g(n) ∈ O(n³).
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- 3. Prove by induction that T(n) = 2T (n/2) + cn is O(n logn).arrow_forwardProve that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 + .... + (2n + 1) = (n + 1)2 for all n ∈ 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
- Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}arrow_forwardLet f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).arrow_forward5. Let g(n) = log10 (n). Prove that g(n) = (lgn). Please show step by step solution. Show the work.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) = √nT(√n) + n, for n>2; T(2) = 1.arrow_forward2) Prove divisible by 3 for any integer nzo. that n(n²+s) is divisible by 3 for Let n 1 1 (1² +5) = 1(1+5)=(6/ K(K²+5) K+ 1 ((k + 1 ) ² + 5arrow_forwardGiven f(n) ∈ Θ(n), prove that f(n) ∈ O(n²).arrow_forward
- 7. 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_forwardf(n) = O(f(n)g(n)) Indicate whether the below is true or false. Explain your reasoning. For all functions f(n) and g(n):arrow_forwardProve by Induction that for all integers n ≥ 1, n < n2 + 1 .Yes this problem is silly, but still do it by induction! Prove by Induction that for all integers n ≥ 3, 2n < n2 .arrow_forward
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