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**한양대학교 정보보호 및 알고리즘 연구실 2008. 1. 28 이재준 담당교수님 : 박희진 교수님**

Growth of function Chapter 3. Growth of function 한양대학교 정보보호 및 알고리즘 연구실 이재준 담당교수님 : 박희진 교수님

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**Contents of Table Asymptotic notation O-notation Ω-notation Θ-notation**

Asymptotically lager/smaller notation o-notation ω-notation 3. Relationship between this notations

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**Asymptotic notation 1. Asymptotic notation**

- Which one is better algorithm for enough inputs? we are studing asymptotic efficiency of algorithms ( size of the input in the limit, as the size of the inputs increases without bound. )

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**g(n) is asymptotically upper bound for f(n)**

1. Asymptotic notation O-notation Asymptotically upper bound - Definition for all n, n ≥ n0 , 0 ≤ f(n) ≤ c g(n), the function f(n) is smaller or equal to g(n) to within constant factor. g(n) is asymptotically upper bound for f(n)

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**O-notation 1. Asymptotic notation - Definition**

O(g(n)) = { f(n) : there exist positive constant c and n0 such that 0 ≤ f(n) ≤ c g(n) for all n ≥ n0 } f(n) cg(n) n0 n

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**O-notation 1. Asymptotic notation**

- f(n) = O(g(n)) means f(n) ∈ O(g(n)) means f(n) = θ(g(n)) implies f(n) = O(g(n)). since θ-notation stronger notation than O-notation O(g(n)) f(n)

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**O-notation Example 1. Asymptotic notation , ( )**

, ( ) Use the definition of O-notation to prove the following property.

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**O-notation Solution to example**

1. Asymptotic notation O-notation Solution to example To show that , we need to find c and n0 For maximize value of , we make n0 = 1 Thus, c=a+\b\ and, n0 = 1 (because c is positive contant)

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**g(n) is asymptotically lower bound for f(n)**

1. Asymptotic notation Ω-notation Asymptotically lower bound - Definition for all n, n ≥ n0 , 0≤ c g(n) ≤ f(n) , the function f(n) is larger or equal to g(n) to within constant factor. g(n) is asymptotically lower bound for f(n)

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**Ω-notation 1. Asymptotic notation - Definition**

Ω (g(n)) = { f(n) : there exist positive constant c and n0 such that 0 ≤ c g(n) ≤ f(n) for all n ≥ n0 } n0 cg(n) f(n) n

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**g(n) is asymptotically tight bound for f(n)**

1. Asymptotic notation θ-notation Asymptotically tight bound - Definition for all n, n ≥ n0 , 0≤ c1g(n) ≤ f(n) ≤ c2g(n), the function f(n) is equal to g(n) to within constant factor. g(n) is asymptotically tight bound for f(n)

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**θ-notation - Definition 1. Asymptotic notation**

θ(g(n)) = { f(n) : there exist positive constant c1, c2 and n0 such that 0 ≤ c1 g(n) ≤ f(n) ≤ c2 g(n) for all n ≥ n0 } c1g(n) n0 f(n) c2g(n) n

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**θ-notation 1. Asymptotic notation**

- f(n) = θ(g(n)) means f(n) ∈ Θ(g(n)) - Definition of θ(g(n)) requires that every member f(n) ∈ Θ(g(n)) be asymptotically nonnegative (=asymptotically positive) θ(g(n)) f(n) f(n) n0

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**θ-notation Example 1. Asymptotic notation ,**

Use the definition of θ-notation to prove the following property.

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**θ-notation Solution to example**

1. Asymptotic notation θ-notation Solution to example When n0=6 then c1 ≤ 0 c1 need to be positive constant so n0=6 is wrong. When n0=1 then ≤ 1/2 (n0=1) When n0=7 then c1 ≤ 1/14 (n0=7) thus, (n0=7)

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**θ-notation Example 1. Asymptotic notation ,**

Use the definition of θ-notation to prove the following property.

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**θ-notation Solution to example**

1. Asymptotic notation θ-notation Solution to example To show that , we need to find c1 , c2 and n0 so, the more value of n is larger, the more value of is smaller So, value of will be smaller than 6. Thus,

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**Theorem 3.1 Relationship 1. Asymptotic notation**

For any two functions f(n) and g(n), we have f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)). θ(g(n)) O(g(n)) Ω(g(n))

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**equal sign to mean set membership**

1. Asymptotic notation Asymptotic notation in equations and inequalities - Stands alone on the right-hand side of an equation equal sign to mean set membership n ∈ O(n2)

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**interpret is as standing for some anonymous function**

1. Asymptotic notation Asymptotic notation in equations and inequalities - Asymptotic notation appears in a formula interpret is as standing for some anonymous function f(n) ∈ Θ(n)

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**Eliminate inessential detail and clutter**

1. Asymptotic notation Asymptotic notation in equations and inequalities - Asymptotic notation appears in a formula There is no point in specifying all lower-order terms exactly Eliminate inessential detail and clutter

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**Number of anonymous functions Number of times the asymptotic notation**

Asymptotic notation in equations and inequalities - Number of anonymous functions Number of anonymous functions Number of times the asymptotic notation

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**Asymptotic notation in equations and inequalities **

- Asymptotic notation appears in left-hand side of an equation right-hand side of an equation provides a coarser level of detail than the left-hand side. There is a way to choose the anonymous functions on the right of the equal sign to make the equation valid. <수학> 하나의 위상(位相) 공간의 위상에 있어서 그 개집합(開集合)이 그 공간의 다른 특정한 위상을 가진 개집합에 포함되어 있는. 이 말은 , 등호의 왼쪽에서 anonymous function을 골라냄으로써, 등호를 valid 하게 할수 있습니다.

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**Asymptotic notation in equations and inequalities **

- A number of such relationships can be chained together f(n) ∈ Θ(n) h(n) ∈ Θ (n2) <수학> 하나의 위상(位相) 공간의 위상에 있어서 그 개집합(開集合)이 그 공간의 다른 특정한 위상을 가진 개집합에 포함되어 있는. g(n) ∈ Θ(n)

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**f(n) is asymptotically smaller than g(n)**

2. Asymptotically larger/smaller notation o-notation Asymptotically smaller - Definition for all n, n ≥ n0 , any positive constant c, c>0, 0 ≤ f(n)< cg(n), the function f(n) is smaller to g(n) to within constant factor. f(n) is asymptotically smaller than g(n)

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**o-notation 2. Asymptotically larger/smaller notation - Definition**

o(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n0 >0 such that 0 ≤ f(n) < cg(n) for all n ≥ n0 } f(n) o(g(n)) n0

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**o-notation 2. Asymptotically larger/smaller notation**

- We use o-notation to denote an upper bound that is not asymptotically tight. - this limit shows the function f(n) becomes insignificant relative to g(n) as n approaches infinity

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**o-notation Example 1. Asymptotic notation**

Use the definition of o-notation to prove the following property. f(n) = o(n2)

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**o-notation Solution to example**

2. Asymptotically larger/smaller notation o-notation Solution to example 2n = o(n2) If f(n) = o((g(n)) then

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**f(n) is asymptotically larger than g(n)**

2. Asymptotically larger/smaller notation ω -notation Asymptotically larger - Definition for all n, n ≥ n0 , any positive constant c, c>0, 0 ≤ cg(n) < f(n), the function f(n) is larger to g(n) to within constant factor. f(n) is asymptotically larger than g(n)

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**ω-notation 2. Asymptotically larger/smaller notation - Definition**

ω(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n0 >0 such that 0 ≤ cg(n) < f(n) for all n ≥ n0 } f(n) ω(g(n))

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**ω -notation 2. Asymptotically larger/smaller notation**

- We use ω-notation to denote a lower bound that is not asymptotically tight. - f(n) ∈ ω(g(n)) if and only if g(n) ∈ o(f(n)). - if the limit exists. That is, f(n) becomes arbitrarily large relative to g(n) as n approaches infinity. Arbitrarily : 독단적으로, 제멋대로, 마음대로

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**ω -notation Example 2. Asymptotically larger/smaller notation**

Use the definition of o-notation to prove the following property. f(n) = ω(n)

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**ω -notation Solution to example**

2. Asymptotically larger/smaller notation ω -notation Solution to example =ω(n) If f(n) = ω((g(n)) then

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**3. Relationship between this notations**

f(n) O(g(n)) o(g(n)) ω(g(n)) Ω(g(n)) θ(g(n)) O(g(n)) Ω(g(n)) ω(g(n)) o(g(n))

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**Transitivity Transpose symmetry 3. Relationship between this notations**

f(n) = Θ(g(n)) and g(n) = Θ(h(n)) imply f(n) = Θ(h(n)) , f(n) = O(g(n)) and g(n) = O(h(n)) imply f(n) = O(h(n)) , f(n) = Ω(g(n)) and g(n) = Ω(h(n)) imply f(n) = Ω(h(n)) , f(n) = o(g(n)) and g(n) = o(h(n)) imply f(n) = o(h(n)) , f(n) = ω(g(n)) and g(n) = ω(h(n)) imply f(n) = ω(h(n)). f(n) O(g(n)) o(g(n)) ω(g(n)) Ω(g(n)) f(n) = O(g(n)) if and only if g(n) = Ω(f(n)), f(n) = o(g(n)) if and only if g(n) = ω(f(n)).

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**Reflexivity Symmetry f(n) = Θ(f(n)) f(n) = O(f(n)) f(n) = Ω(f(n))**

3. Relationship between this notations Reflexivity f(n) = Θ(f(n)) f(n) = O(f(n)) f(n) = Ω(f(n)) Symmetry f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n)). θ(g(n)) O(g(n)) Ω(g(n))

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4. Question and Answer Q&A

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