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 31.2, Problem 3E
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Chapter 31 Solutions
Introduction to Algorithms
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- Show that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).arrow_forwardGiven 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).arrow_forwardLet f(n) and g(n) be asymptotically positive functions. Prove or disprove following. f(n) + g(n) = q(min(f(n), g(n))).arrow_forward
- Find f(n) and big Oarrow_forwardLet f (f(n) and g(n)) be asymptotically nonnegative functions. Using the basic definition of Θ notation, prove that max(f(n), g(n)) = Θ(f(n) + g(n)),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_forward
- Let f(n) and g(n) be asymptotically nonnegative increasing functions. Prove: (f(n) + g(n))/2 = ⇥(max{f(n), g(n)}), using the definition of ⇥ .arrow_forward3. Prove by induction that T(n) = 2T (n/2) + cn is O(n logn).arrow_forwardDetermine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)arrow_forward
- Let f(n) = n2 and g(n) = 3n2-6n+ 4. Show that g(n) e(f(n)) by showing that there exist positive constants no, C1, and ez such that cig(n) < f(n) < o29(n) for all n 2 no-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_forwardLet, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.arrow_forward
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