6.mod 4-6 graphs

7/18/2019 6.mod 4-6 Graphs

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DATA STRUCTURES ANDALGORITHMS

MODULE 4

GRAPHS

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Cities connected through airlines

Delhi

Bangalo

re

kochi

chennai

Bombay Calcutta

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Cities connected through airlines Use trees

With Delhi as root node Delhi

Bangalo

re

kochi

chennai

Bombay Calcutta

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Cities connected through airlineswhich Data structure

Delhi

Bangalo

re

kochi

chennai

Bombay Calcutta

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Cities connected through airlinesUse gra!hs

Delhi

Bangalo

re

kochi

chennai

Bombay Calcutta

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Nonlinear data structure "hose data structures in which data elements are organi#ed in any

arbitrary order $i%e% they are not organi#ed in any se&uence' arecalled non(linear data structures%

)

* + ,

- .

+

) *

-,TREE

GRAPH

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Gra!hs

A gra!h G consists o/ two sets0

a set 12 called the set o/ 3ertices $nodes or !oints'

a set 42 called the set o/ edges $or arc' connecting the3ertices in 1

E"a#!le$

1 5 6A2B2C2D7

4 5 6 $A2B' 2 $B2C' 2 $C2D' 2$A2C' 7

A

C B

D

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Digra!hs

Digra!h 5 directed gra!h

A digra!h G consists o/ two sets0

a set 12 called the set o/ 3ertices $nodes or !oints'

a set 42 called the set o/ edges $ordered elements'connecting the 3ertices in 1

4dge $A2b' is not same as edge $B2A'

E"a#!le$

1 5 6A2B2C2D27

4 5 6 $A2B' 2 $A2C' 2 $C2D'2 $B2A' 7

A

C B

D

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%eighted gra!h

A gra!h or a digra!h with all the edges labeled with some weights

A

C B

D

A

C B

D

5 6

2

3

6

5

5

3

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Ad&acent 'ertices

A 3erte8 1i is ad9acent to $or neighbor o/ ' another 3erte8 say 1k2 i/

there is an edge /rom 1i to 1k

Ad9acent 3ertices o/ A are B and C

Ad9acent 3ertices o/ D is C

A

C B

D

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Sel( Loo!

:/ there is an edge whose starting and ending 3ertices are same

$1i2 1i ' is an edge in the gra!h

A

C B

D

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)arallel edges

:/ there is more than one edge between the same !air o/ 3ertices

A

C B

D

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Multigra!h

A gra!h which has either sel/ loo! or !arallel edges or both

A

C B

D

A

C B

D

Self Loop

Parallel Edge

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MULTIGRA)H

A gra!h which has either sel/ loo! or !arallel edges or both

SIM)LE GRA)H

A gra!h which does not ha3e any sel/ loo! or !arallel edges %

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Co#!lete gra!h

A gra!h or digra!h is said to be com!lete i/ each 3erte8 1i is

ad9acent to e3ery other 3erte8 1 9 in G%

"here are edges /rom any 3erte8 to all other 3ertices

A

C B

D

A

C B

D

A

C B

D

A

C B

D

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C*cle

:/ there is a !ath containing one or more edges which starts /roma 3erte8 1i and terminates into the same 3erte8

A

C B

D

A

C B

D

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Ac*clic gra!h

A gra!h without any cycles

A

C B

D

A

C B

D

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Isolated +erte"

A 3erte8 with no edge connected /rom any other 3erte8 to that3erte8

A

C B

D

4

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Degree o( 'erte"

"he number o/ edges connected with 3erte8 1i

Denoted by degree$1i '

A

C B

D

4

Degree(A) =3 Degree(B) =1 Degree(C) =2 Degree(D) =2 Degree(E) = 0

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Degree o( 'erte"

:n a digra!h there are two degrees

:ndegree ; <o% o/ edges incident to 1i

=utdegree ; <o o/ edges emanating /rom 1i

InDegree(A) =1OutDegree(A) =2

InDegree(B) =1OutDegree(B) =1

InDegree(C) = 1OutDegree(C) =1

InDegree(D) = 1OutDegree(D) =0

A

C B

D

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)endant 'erte"

:/ indegree$1i' 5) and =utDegree$1i'5>

InDegree(A) =1OutDegree(A) =2

InDegree(B) =1OutDegree(B) =1

InDegree(C) = 1OutDegree(C) =1

InDegree(D) = 1OutDegree(D) =0

A

C B

D

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Connected Gra!h ,Not a digra!h-Connected 3ertices 0 i/ there is a !ath /rom 1i to 1k

Connected Gra!h0 i/ /or e3ery !air o/ distinct 3ertices 1i2 1k in G2

there is a !athA

C B

D

4

A

C B

D

4

A Connected Graph Not a Connected Graph

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Strongl* Connected . %ea/l* connected Gra!hs

Strongly connected0 :n a digra!h2 :/ there is a directed !ath /rom 1i to1k and also /rom 1k to 1i /or e3ery !air o/ distinct 3ertices 1i and 1k

Weakly connected0 :/ a digra!h is not strongly connected but theunderlying gra!h $without direction o/ edges ' is connected%

A

C

D

A

C

D

Stronglyconnected

weakly connected

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Co#!lete Gra!h i/ e3ery node u in G is ad9acent to e3ery other node 3 in G%

i%e% "here are edges /rom any 3erte8 to all other 3ertices%

A com!lete gra!h with n nodes will ha3e n,n0-12 edges

A

C B

D

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Re!resentation o( Gra!hs

)% Set Re!resentation

*% ?inked re!resentation

+% Se&uential $matri8' re!resentation

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03 Set Re!resentation "wo sets are maintained

)% Set o/ 3ertices

*% Set o/ edges 2 which is a subset o/ + @ +

:/ its a weighted gra!h2 then set 4 consist o/ + tu!les $% @ + @ +' Where w5 weight o/ the edge

:ts not !ossible to store !arallel edges because in a set 2 twoidentical elements cannot e8ist

4g0. 1 $G'5 6A2 B2 C2 D7

. 4$G' 56 $A2B'2 $A2C'2 $B2C'2 $C2D'7



A

C B

D

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23 Lin/ed re!resentation use ad&acenc* list

.

"he number o/ lists de!ends on the number o/ 3ertices in the gra!h. "he list contains the ad9acent nodes o/ the header node


A

C B

D

B C 8

A C 8

A B D 8

C 8

A

B

C

D

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53 Se6uential ,#atri"- re!resentation Ad&acenc* Matri" ost use/ul way Uses a matri8 o/ order < @ <2 where < is the number o/ 3ertices

4ntries in the matri8 can be decided as /ollowsa u 3 5 )2 i/ there is an edge /rom U to 1

5> 2 otherwise :n multigra!h2 instead o/ ) and >2 the entries would be the

number o/ edges between two 3ertices :n the case o/ weighted gra!h2 the entries would be the weight o/

each edge 4g0

A

C B

D

A 7 C D

A > ) ) >

7 ) > ) >

C ) ) > )

D > > ) >

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53 Se6uential ,#atri"- re!resentation Ad&acenc* Matri" ost use/ul way Uses a matri8 o/ order < @ <2 where < is the number o/ 3ertices

4ntries in the matri8 can be decided as /ollowsa u 3 5 )2 i/ there is an edge /rom U to 1

5> 2 otherwise :n multigra!h2 instead o/ ) and >2 the entries would be the

number o/ edges between two 3ertices

:n the case o/ weighted gra!h2 the entries would be the weight o/

each edge 4g0

A

C B

D

A 7 C D

A > ) ) >

7 ) > ) >

C ) ) > )

D > > ) )

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Ad&acenc* Matri"

Dra89ac/s Dicult to insert and delete 3ertices

:/ number o/ edges is small ; memory wasted $s!arse matri8'

Ad'antages easy to mani!ulate

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O!erations in a gra!h

Se6uential re!resentation ,Ad&acenc* Matri"-

Inserting in a gra!h1erte8 ; insert row and column /or that 3erte8 make

necessary changes

4dge ; !lace 0 at a!!ro!riate !osition in ad9acency matri8

Deleting /rom a gra!h1erte8 ( delete row and column /or that 3erte8 make

necessary changes4dge ; !lace : at a!!ro!riate !osition in ad9acency matri8

Tra'ersing a gra!h7;S ; breadth Erst searchD;S ; de!th Erst search

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Tra'erse a Gra!h

Tra'ersing 'isiting all 'ertices e"actl* once

)% Start /rom a gi3en node

*% 1isit all nodes reachable /rom that starting node

"wo methods used0

53 7;S ; breadth Erst search

Similar to le3el by le3el tra3ersal o/ a tree Any 3erte8 in the le3el : will be 3isited only a/ter the 3isit o/ all

3ertices in its !receding le3el ie% i() akes use o/ a &ueue and a <ag to denote whether a node is

3isited or not

43 D;S ; de!th Erst search Similar to inorder tra3ersal o/ a binary tree Starting /rom a gi3en node2 this tra3ersal 3isits all the nodes u!to

the dee!est le3el and so on% akes use o/ a stac/ and a <ag to denote whether a node is

3isited or not

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Tra'erse a Gra!h

03 7;S ; breadth Erst search

Similar to le3el by le3el tra3ersal o/ a tree

4g0 A C 7 E D ; G

23 D;S ; de!th Erst search

Starting /rom a gi3en node2this tra3ersal 3isits all the nodes u!to the dee!est le3el

and so on%

4g0 A C D G 7 ; E

A

C B

D

4

G

F

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BREADTH -FIRST SEARCH

Similar to le3el by le3el tra3ersal o/ a tree Any 3erte8 in the le3el : will be 3isited only a/ter the 3isit o/

all 3ertices in its !receding le3el ie% i() akes use o/ a 6ueue and a <ag to denote whether a node

is 3isited or not

B dth Fi t S h

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Breadth First Search

%

A

flag=0

B

flag=0

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

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%

A

flag=1

B

flag=0

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

front

Start at any node, say A

Enqueue the node

Change ts flag to !sted

A

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%

A

flag=1

B

flag=1

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

front

"e#o!e the front node, $%

Dsplay $

Add all the negh&ours of $

to the rear of the queue and

'hange ther flag%

B$=A

dsplay

A

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=0

F

flag=0

D

flag=0

G

flag=0

front



'hange ther flag%

B C$=A

dsplay

A

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

front



'hange ther flag%

B C E$=A

dsplay

A

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

front

$o #ore negh&ours

re#anng B C E$=A

dsplay

A

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

front

"e#o!e the ne(t front node, $%

Dsplay $

But no un!sted negh&our for $ C E$=B

dsplay

A B

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

front

$o #ore negh&ours fo node B C E$=B

dsplay

A B

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=0

front


Dsplay $



'hange ther flag%

E D$=C

dsplay

A B C

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=0

front

$o #ore negh&ours E D$=C

dsplay

A B C

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=1

front


Dsplay $



'hange ther flag%

D G$=E

dsplay

A B C E

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=1

front

$o #ore negh&oursD G$=E

dsplay

A B C E

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

front


Dsplay $



'hange ther flag%

G F$=D

dsplay

A B C E D

d h i h

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

front

$o #ore negh&ours%G F$=D

dsplay

A B C E D

B d h Fi S h

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

front


Dsplay $

$o negh&ours

F$=G

dsplay

A B C E D G

B dth Fi t S h

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

front


Dsplay $

$o negh&ours

$=F

dsplay

A B C E D G F

B dth Fi t S h

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%

A

flag=0

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

front

)ueue e#pty

*ra!ersal 'o#plete

$=F

Fnal result

A B C E D G F

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7readth ;irst Search ,7;S-

)% :nitialise all nodes by setting their ag to not 'isited $>'*% Put the starting node into the 6ueue and change its ag to 'isited

$)'

+% while $&ueue not em!ty'03 re#o'e (ront node2 < /rom &ueue23 !rocess < $dis!lay <'+% add all the neigh9ours o/ < to the rear o/ the &ueue set

<ag='isited,% end while

-% sto!

Algorith# 7;S ,start 'erte"-

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7readth ;irst Search ,7;S-)% :nitialise all nodes by settingtheir ag to not 'isited $>'

*% Put the starting node into the6ueue and change its ag to 'isited $)'

+% while $&ueue not em!ty'

03 re#o'e (ront node2 < /rom&ueue

23 !rocess < $dis!lay <'

+% add all the neigh9ours o/ <

to the rear o/ the &ueue set <ag='isited

,% end while

-% sto!

Algorith# 7;S ,start>'erte"-

)% i5)

*% while $iI5n'

)% set 3istedJagKiL5>

*% iMM

+% endwhile

,% set 3isitedJagKstartJ3erte8L5)

-% en&ueue$startJ3erte8'

.% while$rearN5>'

)% 85de&ueue$'

*% dis!lay 8

+% set i5)

,% while$iI5n'

)% i/ $ $aK8LKiL5)' $3isitedKiL5>''

)% 3isitedKiL5)

*% en&ueue$i'*% endi/

-% endwhile

O% endwhile

% sto!

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DEPTH -FIRST SEARCH

Similar to inorder tra3ersal o/ a binary tree Starting /rom a gi3en node2 this tra3ersal 3isits all the nodes u!to

the dee!est le3el and so on% akes use o/ a stac/ and a <ag to denote whether a node is

3isited or not

De!th First Search

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De!th First Search

%

A

flag=0

B

flag=0

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

De!th First Search

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De!th First Search

%

A

flag=1

B

flag=0

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+Start th any node, say A

Change ts flag

Push t to sta'+

A

De!th First Search

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e! s Sea c

%

A

flag=1

B

flag=0

C

flag=0

E

flag=0

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+Pop the top node, $

Dsplay $

Push nto sta'+ all the negh&ours

of $ and 'hange ther state

A

$=A

De!th First Search

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!

%

A

flag=1

B

flag=0

C

flag=1

E

flag=0

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+CPush nto sta'+ all the negh&ours


A

De!th First Search

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!

%

A

flag=1

B

flag=0

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+E CPush nto sta'+ all the negh&ours


A

De!th First Search

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!

%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+B E CPush nto sta'+ all the negh&ours


A

De!th First Search

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!

%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+B E C-sted all negh&ours

A

De!th First Search

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!

%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+E C

A B

Pop the top node, $

Dsplay $

Push nto sta'+ all the

negh&ours of $ and 'hange

ther state

$=B

De!th First Search

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!

%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+E C

A B

$o un!sted negh&ours

$=B

De!th First Search

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!

%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=0

dsplay

top

sta'+C

A B E

Pop the top node, $

Dsplay $



$=E

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=1

dsplay

top

sta'+G C

A B E


of $ and 'hange ther state$=E

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=1

dsplay

top

sta'+G C

A B E

$o #ore negh&ours

$=E

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=1

dsplay

top

sta'+C

A B E G

Pop the top node, $

Dsplay $

But no negh&ours

$=G

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=1

dsplay

top

sta'+C

A B E G

$o #ore negh&ours

$=G

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=0

G

flag=1

dsplay

top

sta'+

A B E G C

Pop the top node, $

Dsplay $



$=C

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=1

dsplay

top

sta'+D

A B E G C


of $ and 'hange ther state $=C

De!th First Search

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%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=1

dsplay

top

sta'+D

A B E G C

$o #ore negh&ours

$=C

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=0

D

flag=1

G

flag=1

dsplay

top

sta'+

A B E G C D

Pop the top node, $

Dsplay $



$=D

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

dsplay

top

sta'+F

A B E G C D


of $ and 'hange ther state$=D

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

dsplay

top

sta'+F

A B E G C D

$o #ore negh&ours

$=D

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

dsplay

top

sta'+

A B E G C D F

Pop the top node, $

Dsplay $

But no negh&ours%

$=F

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

dsplay

top

sta'+

A B E G C D F

$o #ore negh&ours

$=F

De!th First Search

7/18/2019 6.mod 4-6 Graphs


%

A

flag=1

B

flag=1

C

flag=1

E

flag=1

F

flag=1

D

flag=1

G

flag=1

Fnal result

top

sta'+

A B E G C D F

Sta'+ e#pty

*ra!ersal 'o#plete $=F

De!th ;irst Search ,D;S-

7/18/2019 6.mod 4-6 Graphs


! , -

)% initialise all nodes by setting their ag to not 'isited $>'*% !ush the starting node into the stac/ and change its ag to

'isited $)'

+% while $stac/ not em!ty'03 re#o'e to! node2 < /rom stack23 !rocess < $dis!lay <'+% add all the neigh9ours o/ < to the to! o/ the stack set

<ag='isited,% end while

-% sto!

De!th ;irst Search,D;S-

Algorithm DFS $startJ3erte8'

)% i5)

* while $iI5n'

7/18/2019 6.mod 4-6 Graphs


,D;S-

)% initialise all nodes by settingtheir ag to not 'isited $>'

*% !ush the starting node into thestac/ and change its ag to 'isited $)'

+% while $stac/ not em!ty'03 re#o'e to! node2 <

/rom stack

23 !rocess < $dis!lay <'

+% add all the neigh9ours o/ < to the to! o/ the

stack set <ag='isited,% end while

-% sto!

*% while $iI5n'

)% set 3istedJagKiL5>

*% iMM+% endwhile

,% set 3isitedJagKstartJ3erte8L5)

-% !ush$startJ3erte8'

.% while$to!N5>'

)% 85!o!$'

*% dis!lay 8+% set i5)

,% while$iI5n'

)% i/ $ $aK8LKiL5)' $3isitedKiL5>' '

)% 3isitedKiL5)

*% !ush$i'

*% endi/

-% endwhile

O% endwhile

% sto!

Co#!are algorith#s

7/18/2019 6.mod 4-6 Graphs


7;S

)% initialise all nodes by setting theirag to not 3isited $>'

*% !ut the starting node into the 6ueue and change its ag to 3isited $)'

+% while $6ueue not em!ty'

)% remo3e (ront node2 < /rom&ueue

*% !rocess < $dis!lay <'

+% add all the neighbours o/ < to therear o/ the 6ueue setag53isited

,% end while-% sto!

D;S

)% initialise all nodes by setting theirag to not 3isited $>'

*% !ush starting node into the stac/

and change its ag to 3isited $)'+% while $stac/ not em!ty'

)% remo3e to! node2 < /rom stack

*% !rocess < $dis!lay <'

+% add all the neighbours o/ < to theto! o/ the stac/ set

ag53isited,% end while

-% sto!

6.mod 4-6 graphs

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