8 1 bigoh tủ tài liệu training pdf

Asymptotic Analysis Big-‐Oh: Basic Examples Design and Analysis of Algorithms I... Example #1then Claim : if Proof : Choose Need to show that We have, for every and Nextcore AI -Gopal Sh

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

then Claim : if

Proof : Choose

Need to show that

We have, for every

and

Nextcore AI -Gopal Shangari

Claim : for every

Example #2

is not

Proof : by contradiction Suppose

Then there exist constants such that

But then [cancelling from both sides]:

Which is clearly False [contradict i o n ]

Nextcore AI -Gopal Shangari

Nextcore AI -Gopal Shangari

Asymptotic Analysis

Big Oh: Relatives (Omega & Theta)

Design and Analysis

of Algorithms I

OMEGA NOTATION

Definition :

If and only if there exist

constants such that

Picture

Nextcore AI -Gopal Shangari

Theta Notation

if and only if

Defini4on :

and

Equivalent : there exist constants such that

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WHICH IS TRUE? CHECK ALL

LIMLE8OH NOTA4ON

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

Nextcore AI -Gopal Shangari

Claim :

Proof : need to pick constants

Note :

So if we choose

Q.E.D

such that

then (*) holds

Claim :

Proof : by contradiction If

constants such that

then there exist

But then [cancelling ]

Which is certainly false

Q.E.D

EXAMPLE #2

Nextcore AI -Gopal Shangari

Example #3

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

g(nti,

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EXAMPLE #3 (CON( NUEDTI

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Proof :

For every n, we have

And

Thus

Q.E.D