Fm monotonically decreasing if m n implies fm fn fm

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Unformatted text preview: g( n ) = Θ( f ( n )) • Transpose Symmetry f ( n ) = O ( g ( n )) ⇔ g ( n ) = Ω ( f ( n )) f ( n ) = o( g ( n ) ) ⇔ g ( n ) = ω ( f ( n ) ) NCKU IIM NCKU 資料結構 Chapter 3 資料結構 12 Properties of Asymptotic Notation • Function relationship f (n) = Θ( g (n)) f (n) = Ο( g (n)) f (n) = Ω( g (n)) Number relationship a=b a≤b f (n) = o( g (n)) a≥b a<b f (n) = ω ( g (n)) a>b • However, trichotomy does not hold for functions f (n) = n, g (n) = n1+sin n f (n) ≠ Θ( g (n)), f (n) ≠ Ο( g (n)), f (n) ≠ Ω( g (n)) NCKU IIM NCKU 資料結構 Chapter 3 資料結構 13 Standard Notations and Common Functions • Monotonicity: a function f(n) is – – – – monotonically increasing if m ≤ n implies f(m) ≤ f(n). f(m) monotonically decreasing if m ≤ n implies f(m) ≥ f(n). f(m) strictly increasing if m < n implies f(m) < f(n). f(m) strictly decreasing if m > n implies f(m) > f(n). f(m) • Floor & Ceiling: real number x, r, integers n,a,b; r≥0,a>0,b>0 x − 1 < ⎣x ⎦ ≤ x ≤ ⎡x ⎤ < x + 1 ⎡n / 2 ⎤ + ⎣n / 2 ⎦ = n ⎡⎡r / a ⎤ / b ⎤ = ⎡r / ab ⎤ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢⎣ ⎥ ⎦ ⎣ ⎦ ⎣⎢r / a ⎥ /b⎦ = ⎢r / ab⎥ ⎡a / b ⎤ ≤ ( a + ( b − 1 )) / b ⎣a / b...
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