8 1 bigoh tủ tài liệu training pdf

Asymptotic
Analysis
Big-‐Oh: Definition
Design and Analysis
of Algorithms I


Nextcore AI Gopal
Shangari


Nextcore AI Gopal
Shangari


Asymptotic
Analysis
Big-‐Oh: Basic
Examples
Design and Analysis
of Algorithms I


Example #1
Claim : if

Proof : Choose
Need to show that
We have, for every

then

and

Nextcore AI Gopal
Shangari


Example #2
Claim : for every

is not

Proof : by contradiction. Suppose
Then there exist constants

But then [cancelling

such that

from both sides]:

Which is clearly False [contradict i o n ]

Nextcore AI Gopal
Shangari


Asymptotic
Analysis
Big Oh: Relatives

(Omega & Theta)
Design and Analysis
of Algorithms I
Nextcore AI Gopal Shangari


OMEGA NOTATION
Definition :
If and only if there exist
constants
such that

Picture

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Theta Notation
Defini4on :

if and only if

and

Equivalent : there exist constants

such that

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WHICH IS TRUE? CHECK ALL
Let


LIMLE8OHNOTA4ON
Defini4on :
if and only if for all
constants c>0, there exists a constant
such that

Exercise :


Where Does Notation Come From?
•


Asymptotic
Analysis
Additional
Examples
Design and Analysis
of Algorithms I


EXAMPLE #1
Claim :
Proof : need to pick constants

Note :
So if we choose

such that

then (*) holds.

Q.E.D
Nextcore AI Gopal Shangari


EXAMPLE #2
Claim :
Proof : by contradiction. If
constants
such that

But then [cancelling

then there exist

]

Which is certainly false.

Q.E.D
Nextcore AI Gopal Shangari


Example #3

Claim : for every pair of (posi( veti func( ons f(nti,
g(nti,

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Shangari


EXAMPLE #3 (CON( NUEDTI

Proof :
For every n, we have
And

Thus

Q.E.D

Nextcore AI Gopal
Shangari