07a-DepthAndBreathSearch

8/13/2019 07a-DepthAndBreathSearch

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Dr. Ahmad R. Hadaegh

A.R. Hadaegh National University Page 1

Searching

through the

Graphs


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One limitation of the trees is that the trees are hierarchal in

nature. They only represent relations of hierarchal types such as

relations between parents and child

A generalization of tree, called GRAP, is a data structure in

which these limitations are lifted

A graph is a collection of !ertices "nodes# and connectionbetween them called edges

A simple graph G= (V,E)consists of a nonempty set of Vof

!ertices and possibly empty setEof edges where each edge is aconnection between two !ertices

The number of !ertices and edges of a graph are represented as |

V|and |E|,respecti!ely.


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Examples of Simple Graphs

Graph 1 Graph 2

Graph 3 Graph 4


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iagraphs (irecte! Graphs)

A directed graph "diagraph# G(V,E)consists of a non$empty set of V!ertices and a set of edges "arcs# where each

edge is a pair of !ertices from V

The difference between diagraph and simple graphs is that

edge "vi,vj#% "vj,vi#in a simple graph but this is not the

case in a diagraph

ere is an e&ample of a diagraph


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A.R. Hadaegh National University Page $

%ulti&graphs

'n a simple graph, two !ertices can only be (oined by one edge

owe!er, in a multi$graphmore than one edge canconnect the same two!ertices together.

A multi$graph that has anedge going from !erte& vito the same !erte& vi"loop edge# is called

pseudo$graph


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A.R. Hadaegh National University Page '

ath A path from v1to vnis a se)uence of edges edge(v1,v2,

edge(v2, v!, " (vn#1, vn

*+cle 'f v1$ vnand no edge is repeated then it is called a cycle

1 2

3

$

4

'

1 2

3

$

4

'

-n the follo.ing graph, the

e!ges that connects v1to v2an!v2to v!an! v!to v%ma/es a

path from v1to v%0

he path going from v1

to v2an! from v2to v!an! from v!to v1again

ma/es a c+cle0


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Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page

eighte! Graph A graph is called weighted graph if each edge has an

assigned number

*epending on the conte&t in which such a graphs areused, the number assigned to an edge is called itsweight, cost, distance, etc.

*omplete Graph

A graph with n!ertices is called+complete and is denoted as &niffor each pair of distinct !erticese&actly one edge connecting them.

That is each !erte& can be connected to any other !erte& The number of edges in such a graph is-

|V|

|E| = = |V| 5 (|V|&1) 6 27 .hich is 8(|V|2)

2

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8/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 9

Su:&graph A sub$graph 'of graph G=(V, E)is a graph G;=(V;, E;)

such that Vis a subset of VandEis a subset ofE. A

ub$graph induced by !ertices Vis a graph "V, Esuch

that if an edge eis inE, theneis inE

Two !ertices viand vjare called ad(acent if the edge (vi,vjis inE. uch an edge is called in)identwith the !ertices vi

and vj

egree The degree of a !erte& v, deg(vis the number of edges

incident with v. 'f deg(v$*, then vis called isolated

!erte&.


9/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page >

Graph ?epresentation

Graph can be represented"implemented# in se!eral ways.

A simple representation is gi!en byad(acency list that specifies all!ertices ad(acent to each !erte& of agraph

This is called+AR re-resentation

:

c

!

e

f

f

a c !

! e

f

a f

ea :

: !

!a c

gc

a

f

!

:

e

g


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*omparison

/hich representation is better0

Thin1 about searching for an edge

Thin1 about printing all edges

Thin1 about inserting an edge

ow about inserting a !erte&

ow about deleting a !erte&

ow about deleting an edge



Graph raersal

Two systematic method for tra!ersal of graph are depth$first

tra!ersal and breadth$first tra!ersal

Of course since graph may contain cycle, they ha!e different

structure compare to trees

Therefore, tra!ersal algorithms we used for trees cannot be applied

for graphs e&actly the same way and they should be modified to fit

the graph re)uirements

2urther, not all graphs are connected. A graph may contain

isolated trees, isolated graphs or isolated !ertices.

'n a connected graph, we can find a path between any two !ertices

in the graph but this is not the case for unconnected graphs



epth&Airst raersal Bsing ?ecursion

A !erte& vis !isited once then each ad(acent !erte& to vis!isited and so on

'f !erte& vhas no ad(acent !erte& or all of its ad(acent !ertices

ha!e already been !isited, we bac1trac1 to the predecessor of v

The tra!ersal is finished if this !isiting and bac1trac1ing

process leads to the first !erte& where the traversal started

2inally, if there is still some un!isited !erte& in the graph, thetra!ersal continues for one of the un!isited !ertices



AS()

num() = iCC 66 ma/ing ertex isite!

for all ertices u a!acent to

if num(u) is @ 66 it means it is not isite! +etattach e!ge(u) to e!ges

AS(u)D

!epthAirstSearch()

for all ertices

num() = @D 66 initialiing all ertices to unisite!

e!ges = nullD

i = 1D

.hile there is a ertex such that num() is @

AS()D

output e!gesD


15/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 1$

3ote that this algorithm guarantees generating a tree "or a forest"combination of se!eral smaller trees# which includes or spawnso!er all !ertices of the original graph

The tree that meets these condition is called aspanning tree

Also note that an edge is added to edges only if the condition in+i n/m(/ is * is true4 that is if !erte& /reachable from !erte& v

has not been processed

Therefore, certain edges in the original graph do not appear in theresulting tree

The edges included in the tree are calledforward edges"or treeedges# and the ones not included are called back edges

'n the ne&t e&ample, we show bac1 edges with dash lines andforward edges with straight line


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c(2)

a(1)

i

f

:

g

e

h

!

The ad(acent !ertices to aare ), , gandi

Any of the ad(acent !ertices to a

can be chosen randomly as long asit is not !isited already

5ets pic1 ). 6ar1 )as !isited andadd edge a)to the edges

The ad(acent !ertices to )are a, andi

7erte& acannot be chosen because

it is already !isited. 8ut !erte&ori can be chosen

5ets pic1. 6ar1as !isited andadd edge ) to the edges

c(2)

a(1)

i

f(3)

:

g

e

h

!


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19/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 1>

c(2)

a(1)

i(4)

f(3)

:(')

g($)

e

h

!

The ad(acent !ertices togare aand0

7erte& acannot be chosen

because it is already !isited. othe only choice is !erte& 0

6ar1 0as !isited and add edgeg0to the edges

c(2)

a(1)

i(4)

f(3)

:(')

g($)

e()

h

!

The ad(acent !ertices to 0isg,butgis already !isited

/e bac1trac1 togand !erticesad(acent tog are all !isited.

/e bac1trac1 to abut all ad(acent!ertices to aare also !isited

ince ais where we started, wecannot bac1trac1 anymore, so we

pic1 another un!isited !erte& if thereis still one

5ets pic1 eand mar1 eas !isited.


20/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 2@

The only ad(acent !erte& to eis h

6ar1 has !isited and addedge eh to the edges

c(2)

a(1)

i(4)

f(3)

:(')

g($)

e()

h(9)

!

The ad(acent !ertices to hare eandd

7erte& ecannot be chosen

because it is already !isited. othe only choice is !erte& d

6ar1 das !isited and add edge hdto the edges

c(2)

a(1)

i(4)

f(3)

:(')

g($)

e()

h(9)

!(>)



c(2)

a(1)

i(4)

f(3)

:(')

g($)

e()

h(9)

!(>)

The ad(acent !ertices to dis h,buth is already!isited

/e bac1trac1 to handad(acent !ertices to h areall !isited.

/e bac1trac1 to eandad(acent !ertices to eare

also !isited

ince eis where westarted, we cannot

bac1trac1 anymore, so wepic1 another un!isited

!erte& if there is still one

8ut there is no more!erte& left to pic1.Therefore, we are done

c(2)

a(1)

i(4)

f(3)

:(')

g($)

e()

h(9)

!(>)

he final result is



c(2)

a(1)

i

f

:

g

e

h

!

The ad(acent !ertices to aare ),and

Any of the ad(acent !ertices to acan be chosen randomly as longas it is not !isited already

5ets pic1 ). 6ar1 )as !isitedand add edge a )to the edges

c(2)

a(1)

i(3)

f

:

g

e

h

! The only ad(acent !ertices to )is i

6ar1 ias !isited and add edge) ito the edges


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25/35Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 2$

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e

h

! The only ad(acent !erte& to 0

isg

/e mar1gas !isited and add0gto the edges

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e()

h

!

The only ad(acent !erte& togis awhich is already !isited

/e bac1trac1 to 0and the ad(acent!erte& to 0is also !isited

ince we started with 0, we cannotbac1trac1 anymore. o we pic1another un!isited !erte& if theree&ist one

5ets pic1 eand mar1 it as !isited


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Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 2'

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e()

h(9)

! The only ad(acent !erte& to e

is h

/e mar1 has !isited and addehto the edges

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e()

h(9)

!(>) The only ad(acent !erte& to h

is d

/e mar1 das !isited and addhdto the edges


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Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 2

There is no ad(acent!erte& to d

/e bac1trac1 to hand the

ad(acent !erte& to h isalready!isited.

/e bac1trac1 to eand thead(acent !erte& to eis also!isited

ince eis where westarted, we cannot

bac1trac1 anymore, so wepic1 another un!isited

!erte& if there is still one

8ut there is no more!erte& left to pic1.Therefore, we are done

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e()

h(9)

!(>)

c(2)

a(1)

i(3)

f(4)

:($)

g(')

e()

h(9)

!(>)

he final result is


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*omplexit+ of !epth&first search

The comple&ity of depth2irstearch"# is 8(|V| C |E|)because-

'nitializing n/m(vfor each !erte& re)uires |V|steps

D+(vis called deg(vtimes for each v4 that is, once for

each edge of v"to spawn into more calls or to finish the chainof recursi!e calls#, hence, the total number of calls is 2|E|

because each edge is ad(acent to two !ertices

earching for !ertices as re)uired by the statement

hile there is ertex vsuch that n/m(vis @

9an be assumed to re)uire |V|steps in worse case.


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Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 2>

Frea!th&Airst Search

This type of tra!ersal uses )ueue instead of recursion or stac1

:reathAirstSearch()

for all ertices u

num(u) =@D

e!ges = nullD

i = 1D.hile there is a ertex such that num() = @

num() = iCCD

enueue()D 66 pushing into ueue

.hile ueue is not empt+

= !eueue()D 66 popping from the ueue

for all ertices u a!acent to

if num(u) is @

num(u) = iCCD

enueue(u)D

a!! e!ge(u) to e!gesD

8utput e!gesD


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Dr. Ahmad R. HadaeghA.R. Hadaegh National University Page 3@

Original Graph

c

a(1)

i

f

:

g

e

h

! /e pic1 up a !erte&

randomly and mar1 it as

!isited

5ets pic1 !erte& aand

mar1 it as !isited and

place it in the )ueue

Example of Frea!th&Airst Search

a

c

a

i

f

:

g

e

h

!


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c(2)

a(1)

i(3)

f(4)

:

g($)

e

h

!

/e pop a from the )ueue.

The ad(acent !ertices to aareg, , iand )that are not

!isited

/e mar1 them all one by oneand place them in the )ueue"order is not important#

The related edges are added

c

i

f

g

c(2)

a(1)

i(3)

f(4)

:

g($)

e

h

!

/e pop )from the )ueue.

All ad(acent !ertices to )are

!isited already.

o no !erte& goes into the)ueue

i

f

g


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c(2)

a(1)

i(3)

f(4)

:

g($)

e

h

! /e pop ifrom the )ueue.

All ad(acent !ertices to iare !isited already.

f

g

c(2)

a(1)

i(3)

f(4)

:

g($)

e

h

! /e pop from the )ueue.

All ad(acent !ertices toare !isited already.

g


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c(2)

a(1)

i(3)

f(4)

:(')

g($)

e(9)

h()

!(>) /e pop hfrom the )ueue

Ad(acent !ertices to hareeand d

/e mar1 them and placethem in the )ueue

/e also add the relatededges

!

e

c(2)

a(1)

i(3)

f(4)

:(')

g($)

e(9)

h()

!(>) /e pop e

Ad(acent !ertices to eareall !isited already

!


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c(2)

a(1)

i(3)

f(4)

:(')

g($)

e(9)

h()

!(>) /e pop d

Ad(acent !ertices to dareall !isited already

3othing more is left topop.

3o more un!isited !erte&e&ist

Therefore, we are done

he final result is

c(2)

a(1)

i(3)

f(4)

:(')

g($)

e(9)

h()

!(>)

07a-DepthAndBreathSearch

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