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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 Winter '12
 bob
 Big O notation, LG, Order theory, Monotonic function, NCKU IIM

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