[PPT] - graph weight win Rt function its w edges : E on . L Eva PowerPoint Presentation

SLIDE 1 Minimum Cut

t

'

graph

a ( v. E ) win

weight function

w : E →

Rt

n

its

edges

.

SLIDE 2 Minimums Cut S = Eva , Vs } L

T

= V

S

. w CST) q

'

i

:*:*

.

EV

' A

cut

f

Cr is

any

bi partition

( SF )

f

the

vertices

.

(

s ,

T

are both non

empty )

.

E (

s , T) =

set of

edges

crossing

this

cut

SLIDE 3 Minimum Cut 3

✓7

I ✓ 6 ✓ I go 8 8 biz "2 9 vz b 4 A cat

f

Cr

with

the minimum

weight

is called a minimum

cut

.

Problem :

Given

GCV, E. w )

find

a min cut

f

G

.

Applications

?

SLIDE 4 c- V A min

Cst )

cut

is a

cat

f

minimum

weight

9 Where

sand

t

are in me

different

parts

f

the

partition

. ✓7 I

suppose

s =L

13¥

and

✓ I go 8 t= 8

t

= Vg . Vg l OG ✓9 12 r,

wcs.TK#gtsS--Es--Vz3S--

vz•

9£ vz b Y

T

=

{ vi. vz

. . . , Va 3 We can

find

such a cat

by

finding

the maximum

flow

b/w

S and

t

( more

n

this

SLIDE 5 g

Minot of G

( also known as the

global

minced)

is

the

minimum

ver

all Cst )

cats .

If

we use a maximum

flow

subroutine

then

the

runtime

will

be

OCNZMAXFIOW)

(pm

)

.

[

Fastest

MAXFIOW

algorithm

takes

0 (nm )

time

.]

These

algorithms

are

complicated

.

Can

we

avoid

using

flaw

algorithms ?

⑤

SLIDE 6 Stoer-Wagner’s Algorithm 6 .

⑧

Est}

contraction

e ' eye ,

i÷÷÷¥÷

"

lot

6/2*3

Key

bservation

:

The min cut

f

Cr

is

either

the min

Cst )

cut

r
the

minuet

f

G) Est }

SLIDE 7 Stoer-Wagner’s Algorithm

7

# The

above

bservation

leads

to an

iterative

( proto)

algorithm :

( et C 't be the

best

cat

found

so far .

while

IGI

32

i . Find a cat C in

a

which

is

minimum

for

some

pain

,

say

Cst )

. 2 .

If

w ( c) C w CE )

replace

C 't with C . 3 . Contract

( s 't)

and Let

G← GIES,t3

.

SLIDE 8

8

How do

we

find

such

a

cat

?

Inca,

For

some

A- EV

and

a -4 A

⑤(I

Vote,

let

, w ( x, A) =

E

w ( aid )

YEA n

easy ) EE

se

is most-Cited if

w ( n , A ) =

max {

w

Cy

, A) I 94A} .

SLIDE 9 MinCutPhase While If If return g

(G)

A- ← { a } K a is an

arbitrary

vertex

AFV

Let neha most

tightly

connected to A .

IAI

=

IVI -2

s ← se

IAI

=

IVI

I

t ← a

Cc - E ( A , a) A

set of

edges

b/w

n

and A

. A- ←

Au

la ]

Contract

Cst )

SLIDE 10 MinCutPhase While If If return 10

(G)

A- ← { a } K a is an

arbitrary

vertex

AFV

( is

a min Let neha most

tightly

connected to A . (St )

cut of

the

IAI

=

IVI -2 last

two vertices Sc

se

addled

to A .

IAI

=

IVI

l

( mtnyn )

t c- a

Ca -

E ( A , a) A

set of

edges

b/w

n

and A

. A- ← A u { a ]

Contract

Cst )

SLIDE 11 11 ✓7 '

em.vn

v , 8 8 = 4

biz weans , ⇒

'

its::¥¥

WCA ,

Vg ) = 20

vz¥

6 v

I

a= 9

Vz

9 1)

A- { Yo}

2)

w ( { v, } ,

vi. 7=1

, w ( Erol , Va )

and

w ( { Vol , Us ) = 8 A =

{

VG , Vg , ] .

3)

w ( Evo , Vs } , Vt) =L , w ( Eva , ↳ 3 ,# 1=-22

w( Eva, ↳ 3,41=5

SLIDE 12

"

SLIDE 13

÷

.

"

SLIDE 14

i

"

SLIDE 15

÷

.

"

SLIDE 16

"

SLIDE 17

'

÷÷÷÷÷

SLIDE 18 Running Time Mincutphase 13

After

each

contraction

the size of

G

reduces

by 1

.

Let

1- (m ,n ) be the

cost

incurred

during

a

call

to

(G)

.

Total time

OC

n

)

.

SLIDE 19 Running Time Mincutphase 14

After

each

contraction

the size of

G

reduces

by 1

.

Let

1- (m ,n ) be the

cost

incurred

during

a

call

to

(G)

.

Total time

OC

n -1cm , n ) ) . =

0 ( mn

T n' Yn )

.

Finding

the

next most

tightly

connected

vertex

can

be

found

using

a

priority

queue

.

( Extract

Max

,

Increase

key )

thus

Timon )

=

fmtnyn )

SLIDE 20 Proof of Correctness (* IS

( Each

cut

f

the

phase

is a minimum r S

t

cut in the

current

graph

, where s and

t

are the

two

vertices

added

last )

.

The

proof

is

by

induction

.

Suppose

C is

any

arbitrary

Cst )

cut

. we

want to show

wCc*)

E

WCC )

.

SLIDE 21 16

Gr

>

current

graph

so , Tc is the

partition

induced

by C

. + a

a

is mis active ?

S

.

T

suppose

UV is the last

added

vertex

V ' was

added

before

V

V
is

active

if

u

and

v ' are

indifferent parts

in C .

SLIDE 22 17

Suppose

Av

be the

set of

vertices

added

before u . Sve = to Tvc =(AvVU2) Nc

( Ev3UA4nsc

in

. , r , . .

.

. → E- (Ar, r)

T

sve

U Tve

Sc

a

=

Aru Ev }

,

let

C

:

= ( Src ,

Tvc )

.

we

want to show

for

every

active vertex u , " ( Au , v )

E

w ( Cr) .

SLIDE 23

Suppose

Av

be the

set of

vertices

added

before u . ' 8 Sre = Tvc =

Avnsc

← Ann Tc

¥:a

÷

.

E Cat

E- (Arm)

Sc

I

let

Cr

= ( Src ,

Tvc )

.

Note

C 't =

( At

,

)

and

Ct C

. W ( c 't )

Edt )

= w ( e ) .

SLIDE 24

Inductive argument

19

E- (¥ )

.

Base

case !

§

I

°

! First

active

vertex

u

Av

.

tV¥qv3

Wl Av ,

u) = w ( Cr) .

SLIDE 25

Inductive argument

20

Base

case !

§

¥ First

active

vertex

✓

Av

.

tv¥qv3

Wl Av ,

u) =

wlcv)

.

Inductive

step :

suppose

w ( Av , v ) E w ( cu )

holds

upto

the

active

vertex

v .

Let

u be me

first

active

vertex

to

be

added

after

V

SLIDE 26 inLAu : 21 a.

÷.

wCA

= w t w (Auu ) .

SLIDE 27 21

th

÷ .

'

Aun Sc

T

S

C C w ( Au , u ) = w ( Av, u ) t w ( Aal Av , n ) .

But

w

(Ar ,

u )

E

w C Av s v) as u was chosen as the

most

tightly

connected

vertex

w.me .

Au

.

SLIDE 28 21

th

÷ .

'

Aunsc

T

Sc

c w ( Au ,u ) = w ( Abu ) t w ( Aal Av , n ) .

E

w ( Ar , v) t w ( Aal Av, m ) .

But

w (Ar , r ) E W ( cu)

(

inductive

hypothesis)

SLIDE 29

a.÷¥÷÷:÷

.

"

W ( Au , u ) = w ( Av, u ) t w ( Aa l Av , u ) .

E

w ( Ar , v) t w CA n l Ar, m ) .

E

n) Ew

SLIDE 30 21

y

U dy

El Antar ,

u )

a:c. ' Aun Sc T

Sc

c w ( Au , u ) = w ( Abu ) t w ( Aal Av , n ) .

E

w ( Av , v)

t

w ( Aal Av, m ) .

light

blue

edges

in

E ( Aa IA

, u) does

not

contribute

[ ✓

and

nly

to Cee .

SLIDE 31

f

y

ELA

, .u ) 21

an

÷: .

W ( Au , u ) = w ( Av, u ) t w ( Au l Av , n ) .

E

u ( Ar , v)

t

w CA n l Av, m ) .

Ewes ) t

w

E weed

.

wA of

*Is a (c)

O .

SLIDE 32