Fnwgn g n f n c 0 n0 0 such that 0

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Unformatted text preview: e the anonymous function on the rhs to make the equation valid 2n 2 + Θ(n) =Θ(n 2 ) means ∀g(n) ∈ Θ(n), ∃h(n) = Θ(n 2 ) such that 2n 2 + g (n) = h(n) NCKU IIM NCKU 資料結構 Chapter 3 資料結構 10 o-notation & w-notation f(n) is asymptotically smaller than g(n) if f(n)=o(g(n)) o( g (n)) = { f (n) | ∀c > 0, ∃n0 > 0 such that 0 ≤ f (n) < cg (n) ∀n > n0 } f (n) =0 ≈ f (n) < g (n) f ( n) = o ( g ( n)) ⇔ lim n →∞ g (n) f(n) is asymptotically larger than g(n) if f(n)=w(g(n)) ω ( g (n)) = { f (n) | ∀c > 0, ∃n0 > 0 such that 0 ≤ cg (n) < f (n) ∀n > n0 } f ( n) = ω ( g ( n)) ⇔ lim n →∞ Examples: Property: f ( n) =∞ g ( n) 1 2n = o(n 2 ); n 2 ≠ o(n 2 ); 2 ≈ f ( n) > g ( n) 12 1 n = ω (n); n 2 ≠ ω (n 2 ) 2 2 f (n) = o( g (n)) ⇔ g (n) = ω ( f (n)) NCKU IIM NCKU 資料結構 Chapter 3 資料結構 11 Relational Properties • Transitivity f ( n ) = Θ( g( n )) ∧ g( n ) = Θ( h( n )) ⇒ f ( n ) = Θ( h( n )) f ( n ) = O( g( n )) ∧ g( n ) = O( h( n )) ⇒ f ( n ) = O( h( n )) f ( n ) = Ω( g( n )) g( n ) = Ω( h( n )) ⇒ f ( n ) = Ω( h( n )) f ( n ) = o( g( n )) ∧ g( n ) = o( h( n )) ⇒ f ( n ) = o( h( n )) f ( n ) = ω ( g( n )) ∧ g( n ) = ω ( h( n )) ⇒ f ( n ) = ω ( h( n )) • Reflexivity f ( n ) = Θ( f ( n )) f ( n ) = O( f ( n )) f ( n ) = Ω( f ( n )) • Symmetry f ( n ) = Θ( g( n ))...
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