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

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Contents of Table 1.Asymptotic notation O-notation Ω-notation Θ-notation 2.Asymptotically lager/smaller notation o-notation ω-notation 3. Relationship between this notations 2

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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. ) 3 1. Asymptotic notation

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O-notation Asymptotically upper bound - Definition for all n, n n 0, 0 f(n) c g(n), the function f(n) is smaller or equal to g(n) to within constant factor. 4 1. Asymptotic notation g(n) is asymptotically upper bound for f(n)

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O-notation - Definition O(g(n)) = { f(n) : there exist positive constant c and n 0 such that 0 f(n) c g(n) for all n n 0 } 5 1. Asymptotic notation f(n) cg(n) n0n0 n

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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 6 1. Asymptotic notation O(g(n)) f(n)f(n)

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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. 1. Asymptotic notation

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O-notation Solution to example To show that, we need to find c and n 0 For maximize value of, we make n 0 = 1 Thus, c=a+\b\ and, n 0 = 1 (because c is positive contant) 1. Asymptotic notation

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Ω -notation Asymptotically lower bound - Definition for all n, n n 0, 0 c g(n) f(n), the function f(n) is larger or equal to g(n) to within constant factor. 9 1. Asymptotic notation g(n) is asymptotically lower bound for f(n)

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Ω -notation - Definition Ω (g(n)) = { f(n) : there exist positive constant c and n 0 such that 0 c g(n) f(n) for all n n 0 } 10 1. Asymptotic notation n0n0 cg(n) f(n) n

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g(n) is asymptotically tight bound for f(n) θ-notation Asymptotically tight bound - Definition for all n, n n 0, 0 c 1 g(n) f(n) c 2 g(n), the function f(n) is equal to g(n) to within constant factor. 11 1. Asymptotic notation

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θ-notation - Definition θ(g(n)) = { f(n) : there exist positive constant c 1, c 2 and n 0 such that 0 c 1 g(n) f(n) c 2 g(n) for all n n 0 } 12 1. Asymptotic notation c 1 g(n) n0n0 f(n) c 2 g(n) n

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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) 13 1. Asymptotic notation θ(g(n)) f(n)f(n) f(n) n0n0

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θ-notation Example 1. Asymptotic notation, Use the definition of θ-notation to prove the following property., Use the definition of θ-notation to prove the following property.

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

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θ-notation Example 1. Asymptotic notation, Use the definition of θ-notation to prove the following property., Use the definition of θ-notation to prove the following property.

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Thus, θ-notation Solution to example To show that, we need to find c 1, c 2 and n 0 so, 1. Asymptotic notation the more value of n is larger, the more value of is smaller So, value of will be smaller than 6.

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Theorem 3.1 Relationship 18 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) = (g(n)). θ(g(n)) O(g(n))Ω(g(n))

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19 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(n 2 ) equal sign to mean set membership n O(n 2 )

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20 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) interpret is as standing for some anonymous function f(n) Θ(n)

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21 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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22 1. 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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23 1. Asymptotic notation 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.

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24 1. Asymptotic notation Asymptotic notation in equations and inequalities - A number of such relationships can be chained together f(n) Θ(n) g(n) Θ(n) h(n) Θ (n 2 )

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f(n) is asymptotically smaller than g(n) o -notation Asymptotically smaller - Definition for all n, n n 0, any positive constant c, c>0, 0 f(n)< cg(n), the function f(n) is smaller to g(n) to within constant factor. 25 2. Asymptotically larger/smaller notation

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26 2. Asymptotically larger/smaller notation o -notation - Definition o(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n 0 >0 such that 0 f(n) < cg(n) for all n n 0 } f(n) o(g(n)) n0n0

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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 27 2. Asymptotically larger/smaller notation

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o -notation Example 1. Asymptotic notation Use the definition of o-notation to prove the following property. f(n) = o(n 2 ) Use the definition of o-notation to prove the following property. f(n) = o(n 2 )

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

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f(n) is asymptotically larger than g(n) ω -notation Asymptotically larger - Definition for all n, n n 0, any positive constant c, c>0, 0 cg(n) < f(n), the function f(n) is larger to g(n) to within constant factor. 30 2. Asymptotically larger/smaller notation

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31 2. Asymptotically larger/smaller notation ω -notation - Definition ω(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n 0 >0 such that 0 cg(n) < f(n) for all n n 0 } f(n) ω(g(n))

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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. 32 2. Asymptotically larger/smaller notation

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33 2. Asymptotically larger/smaller notation ω -notation Example Use the definition of o-notation to prove the following property. f(n) = ω(n) Use the definition of o-notation to prove the following property. f(n) = ω(n)

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

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35 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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36 3. Relationship between this notations 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) = (g(n)) and g(n) = (h(n)) imply f(n) = (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) = (f(n)), f(n) = o(g(n)) if and only if g(n) = ω(f(n)).

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37 3. Relationship between this notations Reflexivity f(n) = Θ(f(n)) f(n) = O(f(n)) f(n) = (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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38 4. Question and Answer Q&AQ&A

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