Show that 3n3on4 for appropriate c and n0 such that n

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Unformatted text preview: 0, 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n)} Is 3n3 ∈ Θ(n4) ?? How about 22n∈ Θ(2n)?? mp - 7 Comp 122 O-notation For function g(n), we define O(g(n)), big-O of n, as the set: O(g(n)) = {f(n) : ∃ positive constants c and n0, such that ∀ n ≥ n0, we have 0 ≤ f(n) ≤ cg(n) } Intuitively: Set of all functions whose rate of growth is the same as or lower than that of g(n). g(n) is an asymptotic upper bound for f(n). f(n) = Θ (g(n)) ⇒ f(n) = O(g(n)). Θ (g(n)) ⊂ O(g(n)). mp - 8 Comp 122 Examples O(g(n)) = {f(n) : ∃ positive constants c and n0, such that ∀ n ≥ n0, we have 0 ≤ f(n) ≤ cg(n) } Any linear function an + b is in O(n2). How? Show that 3n3=O(n4) for appropriate c and n0. mp - 9 Comp 122 Ω -notation For function g(n), we define Ω (g(n)), big-Omega of n, as the set: Ω (g(n)) = {f(n) : ∃ positive constants c and n0, such that ∀ n ≥ n0, we have 0 ≤ cg(n) ≤ f(n)} Intuitively: Set of all functions whose rate of growth is the same as or higher than that of g(n). g(n) is an asymptotic lower bound for f(n). mp - 10 f(n) = Θ (g(n)) ⇒ f(n) = Ω (g(n)). Θ (g(n)) ⊂ Ω (g(n)). Comp 122 Example Ω (g(n)) = {f(n) : ∃ positive constants c and n0, such that ∀n ≥ n0, we have 0 ≤ cg(n) ≤ f(n)} √n = Ω (lg n). Choose c and n0. mp - 11 Comp 122 Relations Between Θ, O, Ω mp - 12 Comp 122 Relations Between Θ, Ω , O Theorem :: For any two functions g((n) and ff(n), Theorem For any two functions g n) and (n), ff(n) = Θ ((g(n)) iiff (n) = Θ g(n)) ff ff(n) = O((g(n)) and ff(n) = Ω ((g(n)). (n) = O g(n)) and (n) = Ω g(n)). I.e., Θ(g(n)) = O(g(n)) ∩ Ω (g(n)) In practice, asymptotically tight bounds are obtained from asymptotic upper and lower bounds. mp - 13 Comp 122 Running Times “Running time is O(f(n))” ⇒ Worst case is O(f(n)) O(f(n)) bound on the worst-case running time ⇒ O(f(n)) bound on the running time of every input. Θ(f(n)) bound on the worst-case running time ⇒ Θ(f(n)) bound on the running time of every input. “Running time is Ω (f(n))...
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This document was uploaded on 03/26/2014.

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