Projeto e Análise de Algoritmos Notação Assintótica

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧

❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦

Pr♦❥❡t♦ ❡ ❆♥á❧✐s❡ ❞❡ ❆❧❣♦r✐t♠♦s

◆♦t❛çã♦ ❆ss✐♥tót✐❝❛

Pr♦❢✳ ❚✐❛❣♦ ❇✉❛rq✉❡ ❆ss✉♥çã♦ ❞❡ ❈❛r✈❛❧❤♦

❯♥✐❞❛❞❡ ❆❝❛❞ê♠✐❝❛ ❞❡ ●❛r❛♥❤✉♥s ✕ ❯❋❘P❊

❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❈✐ê♥❝✐❛s ❞❛ ❈♦♠♣✉t❛çã♦

✷✽ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✶✽

(2)

❘♦t❡✐r♦

✶

■♥tr♦❞✉çã♦

✷

◆♦t❛çã♦

Θ

✸

◆♦t❛çã♦

O

✹

◆♦t❛çã♦

Ω

✺

◆♦t❛çã♦

o

✻

◆♦t❛çã♦

ω

✼

❈♦♠♣❛r❛çã♦ ❞❡ ❋✉♥çõ❡s

✽

◆♦t❛çã♦ ❡ ❋✉♥çõ❡s ❈♦♠✉♥s

(3)

❊✜❝✐ê♥❝✐❛ ❛ss✐♥tót✐❝❛

• _{❖r❞❡♥❛çã♦ ♣♦r ✐♥s❡rçã♦ é}

Θ(

n

2 )

✳

• _{❖r❞❡♥❛çã♦ ♣♦r ✐♥t❡r❝❛❧❛çã♦ é}

Θ(

n

lg

n

)✳

• _P❛r❛

n

❛ ♣❛rt✐r ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r✱ ❛ ♦r❞❡♥❛çã♦ ♣♦r ✐♥t❡r❝❛❧❛çã♦

s❡rá s❡♠♣r❡ ♠❛✐s rá♣✐❞❛✱

• _{♣♦rq✉❡ ❛ ♦r❞❡♥❛çã♦ ♣♦r ✐♥t❡r❝❛❧❛çã♦ é ♠❛✐s ❡✜❝✐❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡✳}

(4)

(5)

(6)

◆♦t❛çã♦ ❆ss✐♥tót✐❝❛

• _{●❡r❛❧♠❡♥t❡ ✉t✐❧✐③❛❞❛ ❝♦♠}

n

∈

N

✱

N

₌

_{

₀

_,

₁

_,

₂

_{, . . .}}

✳

• _{❯t✐❧✐③❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞♦s ❛❧❣♦r✐t♠♦s✳}

• _{➱ ❛♣❧✐❝❛❞❛ ❛ ❢✉♥çã♦ q✉❡ ❞❡s❝r❡✈❡ ♦ t❡♠♣♦ ❞♦ ❛❧❣♦r✐t♠♦✳}

• _{❊①❡♠♣❧♦✿ ♦ t❡♠♣♦ ❞❛ ♦r❞❡♥❛çã♦ ♣♦r ✐♥s❡rçã♦ é}

an

2 +

bn

+

c

✱

• _{q✉❡ ♣♦❞❡ s❡r ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦✉tr♦s t❡♠♣♦s ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦}

Θ(

n

2 )

.

(7)

(8)

◆♦t❛çã♦

Θ

• _{❆ ♦r❞❡♥❛çã♦ ♣♦r ✐♥s❡rçã♦ é}

T

(

n

) = Θ(

n

2 )

.

Θ(

g

(

n

)) =

{

f

(

n

)

|

❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c1

✱

c2

❡

n0

t❛✐s q✉❡

0 ≤

c1

g

(

n

)

≤

f

(

n

)

≤

c2g

(

n

)

♣❛r❛ t♦❞♦

n

≥

n0}.

• Θ(

g

(

n

))

é ✉♠ ❝♦♥❥✉♥t♦✱ ❡♥tã♦ ❞❡✈❡r✐❛✲s❡ ❡s❝r❡✈❡r ✏

f

(

n

)

∈

Θ(

g

(

n

))

✑ ❡

♥ã♦ ✏

f

(

n

) = Θ(

g

(

n

))

✑✬✱ ♠❛s ❛♠❜❛s ❛s ❢♦r♠❛s sã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛

✐♥❞✐❝❛r ❛ ♠❡s♠❛ ❝♦✐s❛✳

• g

(

n

)

é ✉♠ ❧✐♠✐t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ r❡str✐t♦ ♣❛r❛

f

(

n

)

• f

(

n

)

❡

g

(

n

)

sã♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ♥ã♦ ♥❡❣❛t✐✈♦s ♣❛r❛

n

s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

(9)

❊①❡♠♣❧♦

•

1

2 n

2 −

3 n

= Θ(

n

2 )

✳

• _{➱ ♣r❡❝✐s♦ ❡♥❝♦♥tr❛r}

c1

✱

c2

❡

n0

t❛✐s q✉❡

0 ≤

c1n

2 ≤

1

2 n

2 −

3 n

≤

c2n

2 ,

♣❛r❛

n

≥

n0

✳

• _{❉✐✈✐❞✐♥❞♦ t✉❞♦ ♣♦r}

n

2 _✿

0 ≤

c1

≤

1

2 −

3 n

≤

c2

✱ ♣❛r❛

n

≥

n0

✳

(10)

• _{◗✉❛♥t♦ ♠❛✐♦r}

n

✱ ❛ ❡①♣r❡ssã♦

1

2 −

3 n

t❡♥❞❡ ❛

1 /

2 ✱ ♣♦rt❛♥t♦

1

2 −

3 n

≤

1

2 ✳ P♦❞❡♠♦s ❡s❝♦❧❤❡r

c2

= 1

/

2 .

• _{P❛r❛ ❞❡t❡r♠✐♥❛r}

n

0 ✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ r❡str✐çã♦

1

2 −

3 n0

≥

0 ♦❜t❡♥❞♦

n0

≥

6 ✳

• _P❛r❛

n0

= 6

t❡r❡♠♦s

c1

= 0

✳

• _{❙❡ q✉✐s❡r♠♦s ❡✈✐t❛r ❛♥✉❧❛r ❛ ♠✉❧t✐♣❧✐❝❛çã♦}

c1g

(

n

)

✱ ✐st♦ é✱ ❢❛③❡r

c1

>

0 ✱

♣♦❞❡♠♦s ❢❛③❡r

n0

= 7

♦❜t❡♥❞♦

c1

= 1

/

2 −

3 /

7 = 1

/

14 ✳

• _P❛r❛

c1

= 1

/

14 ✱

c2

= 1

/

2 ❡

n0

= 7

✱ t❡♠♦s

1

2 n

2 −

3 n

= Θ(

n

2 )

.

(11)

❖✉tr♦ ❡①❡♠♣❧♦

• _{P❛r❛ ♠♦str❛r q✉❡}

6 n

3

6 = Θ(

n

2 )

✳

• _{Pr♦✈❛ ♣♦r ❝♦♥tr❛❞✐çã♦✳}

• _{❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠}

c

2 ❡

n

0 t❛✐s q✉❡

6 n

3 ≤

c

2 n

2 .

• _{❙✐♠♣❧✐✜❝❛♥❞♦✿}

n

≤

c

2 /

6 .

• _{❊st❛ ❝♦♥❞✐çã♦ ♥✉♥❝❛ é ✈❡r❞❛❞❡ ♣♦✐s}

c2

é ❝♦♥st❛♥t❡ ❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r

✉♠

n

❝❛❞❛ ✈❡③ ♠❛✐♦r✳

• _{❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡✲s❡ s❡♠♣r❡ ✐❣♥♦r❛r ♦s t❡r♠♦s ❞❡ ♦r❞❡♠}

✐♥❢❡r✐♦r✳

(12)

❆❧❣✉♥s ❝❛s♦s ❣❡r❛✐s

• _{P♦❞❡✲s❡ s❡♠♣r❡ ❡s❝♦❧❤❡r}

c1

✉♠ ♣♦✉❝♦ ♠❡♥♦r q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ♠❛✐s

❛❧t❛ ♦r❞❡♠ ❡

c2

✉♠ ♣♦✉❝♦ ♠❛✐♦r✳

• _❙❡❥❛

f

(

n

) =

an

2 +

bn

+

c

✱

a >

0 ✳

• f

(

n

) = Θ(

n

2 )

✱

c1

=

a/

4 ✱

c2

= 7

a/

4 ❡

n0

= 2

×

max

|b|/a,

p

|c|/a

✳

• _{P❛r❛ q✉❛❧q✉❡r ♣♦❧✐♥ô♠✐♦}

p

(

n

) =

P

d

i

=0

a

i

n

i

✱ ❝♦♠

a

i

s❡♥❞♦ ❝♦♥st❛♥t❡s

❡

a

d

>

0✳

• p

(

n

) = Θ(

n

d

)

✳

• _{◗✉❛❧q✉❡r ❢✉♥çã♦ ❝♦♥st❛♥t❡ é}

Θ(

n

0 )

♦✉

Θ(1)

✳

(13)

(14)

◆♦t❛çã♦

O

✭❖✲❣r❛♥❞❡✱ ❇✐❣✲❖ ❡♠ ■♥❣❧ês✮

• O

(

g

(

n

))

• O

(

g

(

n

))

✱ ✏Ó ❣r❛♥❞❡ ❞❡

g

❞❡

n

✑✳

O

(

g

(

n

)) =

{

f

(

n

)

|

❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c

❡

n

0 t❛✐s q✉❡

0 ≤

f

(

n

)

≤

cg

(

n

)

♣❛r❛ t♦❞♦

n

≥

n0}.

• _❙❡

f

(

n

) = Θ(

g

(

n

))

❡♥tã♦

f

(

n

) =

O

(

g

(

n

))

.

• Θ(

g

(

n

))

⊆

O

(

g

(

n

))

.

• an

2 +

bn

+

c

✱

a >

0 ✱ ❡stá ❡♠

Θ(

n

2 )

❡ t❛♠❜é♠ ❡♠

O

(

n

2 )

✳

• an

+

b

❡stá ❡♠

O

(

n

2 )

✱

c

=

a

+

|b|

❡

n0

= max(1

,

−b/a

)

.

• _{▲✐♠✐t❡ ❛ss✐♥tót✐❝♦ s✉♣❡r✐♦r✳}

• _{P✐♦r ❝❛s♦✳}

(15)

(16)

◆♦t❛çã♦

Ω

✭ô♠❡❣❛✮

• _{▲✐♠✐t❡ ❛ss✐♥tót✐❝♦ ✐♥❢❡r✐♦r✳}

Ω(

g

(

n

)) =

{

f

(

n

)

|

❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c

❡

n

0 t❛✐s q✉❡

0 ≤

cg

(

n

)

≤

f

(

n

)

♣❛r❛ t♦❞♦

n

≥

n0}.

• _{❚❊❖❘❊▼❆✿ ♣❛r❛ q✉❛✐sq✉❡r ❞✉❛s ❢✉♥çõ❡s}

f

(

n

)

❡

g

(

n

)

✱ t❡♠♦s

f

(

n

) = Θ(

g

(

n

))

s❡ ❡ s♦♠❡♥t❡ s❡

f

(

n

) =

O

(

g

(

n

))

❡

f

(

n

) = Ω(

g

(

n

))

.

• an

2 +

bn

c

= Θ(

n

2 )✱

a >

0✱ ✐♠♣❧✐❝❛

an

2 +

bn

c

=

O

(

n

2 )

❡

an

2 +

bn

c

= Ω(

n

2 )

✳

• _{▼❡❧❤♦r ❝❛s♦✳}

• _{❖r❞❡♥❛çã♦ ♣♦r ✐♥s❡rçã♦ é}

Ω(

n

)

❡

O

(

n

2 )

.

(17)

◆♦t❛çã♦ ❛ss✐♥tót✐❝❛ ❡♠ ❡q✉❛çõ❡s

• _{❈❛s♦ ♣❛❞rã♦✿}

f

(

n

) =

O

(

g

(

n

))

s✐❣♥✐✜❝❛

f

(

n

)

∈

O

(

g

(

n

))

❀

• _{❊♠ ❡q✉❛çã♦ r❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦ ❣❡♥ér✐❝❛ s❡♠ ❝♦♥st❛♥t❡s}

❡s♣❡❝✐✜❝❛❞❛s✳

•

2 n

2 + 3

n

+ 1 = 2

n

2 +

f

(

n

)

→

f

(

n

) = 3

n

+ 1

• T

(

n

) = 2

T

(

n/

2) + Θ(

n

) = 2

T

(

n/

2) +

an

+

b

= 2

T

(

n/

2) +

cn.

• _{❉♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❡q✉❛çã♦}

2 n

2 + Θ(

n

) = Θ(

n

2 )

.

• _{❊♥❝❛❞❡❛♥❞♦ r❡❧❛çõ❡s}

2 n

2 + 3

n

+ 1 = 2

n

2 + Θ(

n

)

✱

2 n

2 + Θ(

n

) = Θ(

n

2 )

.

(18)

◆♦t❛çã♦

o

✭ó✲♣❡q✉❡♥♦✮

•

2 n

2 =

O

(

n

2 )

é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❥✉st♦✳

•

2 n

=

O

(

n

2 )

◆➹❖ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❥✉st♦✳

• _{❆ ♥♦t❛çã♦}

o

❞❡✜♥❡ ✉♠ ❧✐♠✐t❡ q✉❡ ◆➹❖ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❥✉st♦✳

o

(

g

(

n

)) =

{

f

(

n

)

|

♣❛r❛ q✉❛❧q✉❡r ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c

❡

n

0 t❛✐s q✉❡

0 ≤

f

(

n

)

≤

cg

(

n

)

♣❛r❛ t♦❞♦

n

≥

n0}.

• _{❊①❡♠♣❧♦✿}

2 n

=

o

(

n

2 )

♠❛s

2 n

2

6 =

o

(

n

2 )

.

• _{◆♦t❛çã♦}

O

✏♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡

c >

0 ✑

• _{◆♦t❛çã♦}

o

✏♣❛r❛ t♦❞❛s ❛s ❝♦♥st❛♥t❡s

c >

0 ✑

lim

n→∞

f

(

n

)

g

(

n

)

= 0

.

(19)

◆♦t❛çã♦

ω

✭ô♠❡❣❛ ♣❡q✉❡♥♦✮

• f

(

n

)

∈

ω

(

g

(

n

))

s❡ ❡ s♦♠❡♥t❡ s❡

g

(

n

)

∈

o

(

f

(

n

))

.

ω

(

g

(

n

)) =

{

f

(

n

)

|

♣❛r❛ q✉❛❧q✉❡r ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c

❡

n0

t❛✐s q✉❡

0 ≤

cg

(

n

)

≤

f

(

n

)

♣❛r❛ t♦❞♦

n

≥

n0}.

• _{❊①❡♠♣❧♦s✿}

n

2 /

2 =

ω

(

n

)

✱ ♠❛s

n

2 /

2

6 =

ω

(

n

2 )

.

lim

n→∞

f

(

n

)

g

(

n

)

=

∞.

(20)

❈♦♠♣❛r❛çã♦ ❞❡ ❋✉♥çõ❡s

(21)

❈♦♠♣❛r❛çã♦ ❞❡ ❋✉♥çõ❡s

(22)

❆♥❛❧♦❣✐❛ ❝♦♠ ♦s ♥ú♠❡r♦s ❘❡❛✐s

(23)

◆♦t❛çã♦ ❡ ❋✉♥çõ❡s ❈♦♠✉♥s

• _{▼♦♥♦t♦♥✐❝✐❞❛❞❡✱}

• _{P✐s♦s ❡ t❡t♦s✱}

• _{❆r✐t♠ét✐❝❛ ♠♦❞✉❧❛s✱}

• _{P♦❧✐♥ô♠✐♦s✱ ❊①♣♦♥❡♥❝✐❛✐s✱ ▲♦❣❛r✐t♠♦s✱ ❋❛t♦r✐❛✐s✱}

• _{■t❡r❛çã♦ ❢✉♥❝✐♦♥❛❧✱ ❛ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦ ✐t❡r❛❞♦✱}

• _{◆ú♠❡r♦ ❞❡ ❋✐❜♦♥❛❝❝✐✳}

(24)

P✐s♦ ❡ t❡t♦

(25)

❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r

(26)

P♦❧✐♥ô♠✐♦s

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

❘❡❢❡rê♥❝✐❛

▲✐✈r♦✿

❚✳❍✳ ❈♦r♠❡♥✱ ❈✳❊✳ ▲❡✐s❡rs♦♥✱ ❘✳▲✳ ❘✐✈❡st✱ ❈✳ ❙t❡✐♥✳ ❆❧❣♦r✐t♠♦s✿ ❚❡♦r✐❛ ❡

Prát✐❝❛✳ ❊❧s❡✈✐❡r✱ ✷✵✶✷✳

✲

◆❡st❛ ❛✉❧❛✿ ❈❛♣ít✉❧♦ ✸✳

◆❡st❛ ❛✉❧❛ ❢♦r❛♠ ❝♦♣✐❛❞♦s ♣❛rt❡ ❞♦ ❧✐✈r♦✳