Growth of function 2008. 1. 28 : 1 Chapter 3. Growth of function.

Growth of function

한양대학교 정보보호 및 알고리즘 연구실

2008. 1. 28이재준

담당교수님 : 박희진 교수님1

Chapter 3. Growth of function

Contents of Table

1. Asymptotic notationO-notationΩ-notationΘ-notation

2. Asymptotically lager/smaller notation o-notation ω-notation

3. Relationship between this notations

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• 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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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.

4


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

• O-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

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f(n)

cg(n)

n0

n

• O-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

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O(g(n))

f(n)

))(())(( ngOng

• O-notation Example

, ( )

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

, ( )

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

bannf )(

)()( 2nOnf

0a


• 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)


cn

b

n

a

cnban

2

2

0

0

2n

b

n

a

20 cnban

cba ||0

• 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.

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

• Ω-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

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n0

cg(n)

f(n)

n

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

• 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.

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• θ-notation - Definition

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

12


c1g(n)

n0

f(n)

c2g(n)

n

• θ-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)

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θ(g(n)) f(n)

f(n)

n0

• θ-notation Example


,

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

,


nnnf 32

1)( 2

)()( 2nnf

• θ-notation Solution to example


221

22

221

3

2

10

32

10

cn

c

ncnnnc

nc

3

2

10 1

When n0=6 then c1 ≤ 0 c1 need to be positive constant so n0=6 is wrong.When n0=7 then c1 ≤ 1/14

(n0=7)14

114

10

1

1

c

c

2

3

2

1c

n

When n0=1 then ≤ 1/2n

3

2

1

2

12 c (n0=1)

thus, (n0=7)2

12 c

14

11 c

• θ-notation Example


,


,


36)( nnf

)()( 2nnf

Thus, Thus,

• θ-notation Solution to example To show that , we need to find c1 , c2 and n0

so,


n

c

n

c 21 6

22

321 6 ncnnc

n

c26

the more value of n is larger, the more value of is smaller

So, value of will be smaller than 6.

n

c2

n

c2

)()( 2nnf

• Theorem 3.1

• Relationship

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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)).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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• Asymptotic notation in equations and inequalities

- Stands alone on the right-hand side of an equation

)( 2nOn equal sign to mean set

membershipn ∈ O(n2)

equal sign to mean set membership

n ∈ O(n2)

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- Asymptotic notation appears in a formula

)(2132 22 nnnn

)(2132 22 nfnnn

interpret is as standing for some anonymous function

f(n) ∈ Θ(n)

interpret is as standing for some anonymous function

f(n) ∈ Θ(n)

)(13 nn

)13)(( nnf

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- Asymptotic notation appears in a formula

There is no point in specifying all lower-order terms exactly

)()2/(2)( nnTnT

Eliminate inessential detail and clutter

Eliminate inessential detail and clutter

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- Number of anonymous functions

Number of anonymous functions

Number of anonymous functions

Number of times the asymptotic

notation

Number of times the asymptotic

notation

)(...)2()1()(1

nOOOiOn

i

)(nO

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- 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.

)()(2 22 nnn

There is a way to choose the anonymous functions on the right of the equal sign to make the equation valid.

There is a way to choose the anonymous functions on the right of the equal sign to make the equation valid.

)()(2 2 ngnfn

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- A number of such relationships can be chained together

)(2132 22 nnnn

)( 2n

f(n) ∈ Θ(n)f(n) ∈ Θ(n)

)(2132 22 nfnnn

g(n) ∈ Θ(n)g(n) ∈ Θ(n)

)()(2 2 nhngn h(n) ∈ Θ (n2)h(n) ∈ Θ (n2)

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

• 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.

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2. Asymptotically larger/smaller notation

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• o-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

• o-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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0)(

)(lim

ng

nf

n

• o-notation Example


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

f(n) = o(n2)


f(n) = o(n2)

nnf 2)(

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• o-notation Solution to example

2n = o(n2)

If f(n) = o((g(n)) then

0)(

)(lim

ng

nf

n

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

• 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.

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• ω-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))

• ω -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.

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)(

)(lim ng

nf

n

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• ω -notation Example


f(n) = ω(n)


f(n) = ω(n)

2)(

2nnf

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• ω -notation Solution to example

=ω(n)

If f(n) = ω((g(n)) then

2n

)(

)(lim ng

nf

n

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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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f(n)

O(g(n))

o(g(n))

ω(g(n))

Ω(g(n))

• Transitivity

• Transpose symmetry

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)) 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

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

Growth of function 2008. 1. 28 : 1 Chapter 3. Growth of function.

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