Graphs Spanning

8/17/2019 Graphs Spanning

1/51

James Tam

Minimum Spanning Trees

•In this section of notes you will learn twoalgorithms for creating a connected graphat minimum cost as well as a method forordering a graph


2/51

James Tam

Minimum Spanning Trees

•Applies to weighted, undirected and connected graph

•Create the minimum number of edges/arcs so that all

nodes/vertices are connected


3/51

James Tam

Example Of A Problem Dealing With MinimumSpanning Trees

http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=22960&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998


4/51

James Tam


http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=22960&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998


5/51

James Tam


$$$

http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=22960&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=21230&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998


6/51

James Tam


http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=17711&size=big&papass=&sort=1&thecat=998


7/51James Tam

Algorithms For Determining The MinimumSpanning Tree

•Prims Algorithm

•!rus"als Algorithm


8/51James Tam

Prim’s Algorithm

primsAlgorithm #$raph g, Tree t, %ode start&

'

Priority(ueue nodes)eft * new Priority(ueue#&+

%ode temp+

%ode neighbor+

for #int i * + i -* g.no%odes #&+ i&

'

g0i1.set2eight * 3+

g0i1.setParent #null&+

nodes)eft.add #g0i1&+

4

start.set2eight #5&+

Pseudo code is based roughly on the algorithm provided by Matthew A. Becker

http://www.andrew.cmu.edu/user/mbecker/


9/51James Tam

Prim’s Algorithm (2

while #nodes)eft.is6mpty#& ** false&

'

temp * nodes)eft.de7ueue8in2eight%ode #&+

t.add #temp&+

for each node 9neighbor: which is ad;acent to temp and is in nodes)eft

'

if #weight #temp, neighbor& - neighbor.get2eight#&&

'

neighbor.setParent #temp&+

neighbor.set2eight #weight #temp, neighbor&&+ 4

4

4

4Pseudo code based roughly on the algorithm provided by Matthew A. Becker

http://www.andrew.cmu.edu/user/mbecker/


10/51James Tam

Example! Fin"ing The Minimum Spanning Tree #nA $raph (Prim’s Algorithm

i

a b

c

d

e

f

g

h

4

6

!

"

#

$%

4

&

$


11/51James Tam

#nitiali%e" &alues For The $raph An" 'ueue

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Graph

Queue

2eight ) ) ) ) ) ) ) )

%ode A * + D E F $ , #

Predecessor

Tree


12/51James Tam

Mar- .A/ As &isite"0 1p"ate eighbors

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 ) ) ) 4 ) ) 2

%ode A * + D E F $ , #

Predecessor 5 A A A

Tree

a

Graph


13/51James Tam

Mar- o"e .#/ As &isite" An" #n5lu"eThe E"ge (A0#

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 ) ) ) 4 ) ) <

%ode A * + D E F $ , I

Predecessor 5 A A A

Tree

a

i

Graph


14/51James Tam

Mar- o"e .F/ As &isiste" An" #n5lu"eThe E"ge (A0F

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 ) ) ) = 2 ) <

%ode A * + D E > $ , I

Predecessor 5 A A F A

Tree

a

i

f

Graph


15/51James Tam

Mar- o"e .$/ As &isiste" An" #n5lu"eThe E"ge (F0$

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 ) 6 7 4 < ) <

%ode A * + D E F $ , I

Predecessor 5 A $ $ A > A

Tree

a

i

f

g

Graph


16/51James Tam

Mar- o"e .D/ As &isiste" An" #n5lu"eThe E"ge (D0$

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 4 ? 7 = < 8 <

%ode A * + @ E > $ , I

Predecessor 5 A D $ $ A > D A

Tree

a

i

f

g

d

Graph


17/51James Tam

Mar- o"e .,/ As &isiste" An" #n5lu"eThe E"ge (D0,

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 4 ? 7 = < <

%ode A * + @ E > $ I

Predecessor 5 A D $ $ A > @ A

Tree

a

i

f

g

d h

Graph


18/51James Tam

Mar- o"e .+/ As &isiste" An" #n5lu"eThe E"ge (+0D

i

a b

c

d

e

f g

h

4

6

!

"

#$ %

4

&

$

Queue

2eight 5 3 = ? 9 = < <

%ode A * C @ E > $ I

Predecessor 5 A @ $ + A > @ A

Tree

a

i

f

g

d h

c

Graph


19/51James Tam

Mar- o"e .E/ As &isiste" An" #n5lu"eThe E"ge (+0E

i

a b

c

d

e

f g

h

4

6

!

"

$ %

4

&

$

Queue

2eight 5 3 = ? B = < <

%ode A * C @ 6 > $ I

Predecessor 5 A @ $ C A > @ A

Tree

a

i

f

g

d h

c

e

Graph


20/51James Tam

Mar- o"e .*/ As &isite" An" #n5lu"eThe E"ge (A0*

i

a b

c

d

e

f g

h

4

6

"

$ %

4

&

$

Tree

a

i

Queue

2eight 5 = ? B = < <

%ode A D C @ 6 > $ I

Predecessor 5 A @ $ C A > @ A

f

g

d h

c

eb

Graph


21/51James Tam

:rus-al’s Algorithm! Des5ription

•Etart out with a 9forest: of unconnected trees #9eFtract: the

nodes from the graph&

'ode

'ode 'ode

'ode

Becomes

http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998


22/51James Tam

:rus-al’s Algorithm! Des5ription

•8a"e the connections between the separate trees #add edges&

which have the lowest cost.

•Connecting two separate nodes #two separate trees& merges

them into one tree

http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998http://www.barrysclipart.com/barrysclipart.com/showphoto.php?photo=20761&papass=&sort=1&thecat=998


23/51James Tam

:rus-al’s Algorithm

public Eet "rus"al #$raph g&

'

Eet combinedEet * new Eet #&+

int edgesAccepted * 5+

Priority(ueue edges)eft * g.sort6dges #&+

6dge used6dge+

%ode source%ode+

%ode destination%ode+


24/51

James Tam

:rus-al’s Algorithm (2

while #edgesAccepted - #g.get%umber%odes#& G &&

'

used6dge * edges)eft.de7ueue8in #&+

source%ode * used6dge.getEource%ode #&+

destination%ode * used6dge.get@estination%ode #&+

if ##source%ode H* destination%ode&

#notCycle#source%ode,destination%ode,combinedEet& ** true&

'

edgesAccepted+

combinedEet.union #source%ode, destination%ode&+

4

4

return combinedEet+

4


25/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Original $raph

(

A

)

*

B

+

,

4

$

&&-

$$

"

% # 4 6

&


26/51


27/51


28/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!First E"ge A""e" (A=D

Priorit; 8

A = *! Weight > 2

+ = D! Weight > 2

D = E! Weight > 2

* = D! Weight > 9

A = +! Weight > 4

D = $! Weight > 4+ ? F! Weight > 6

E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&


29/51


30/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Thir" E"ge A""e" (A=*

Priorit; G $ 2eight *

A G D 2eight * <

+ = D! Weight > 2

D = E! Weight > 2

* = D! Weight > 9

A = +! Weight > 4


E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$


31/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Fourth E"ge A""e" (+=D


A G D 2eight * <

C G @ 2eight * <

D = E! Weight > 2

* = D! Weight > 9

A = +! Weight > 4


E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$

$


32/51


33/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Don’t 1se E"ge (*=D


A G D 2eight * <

C G @ 2eight * <

@ G 6 2eight * <

D G @ 2eight * B

A = +! Weight > 4


E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$

$

"

S-ip$


34/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Don’t 1se E"ge (A=+


A G D 2eight * <

C G @ 2eight * <

@ G 6 2eight * <

D G @ 2eight * B

A G C 2eight * =


E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$

$

"

S-ip$

S-ip

4

E l T Of : - l’ Al i h


35/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!Sixth E"ge A""e" (D=$


A G D 2eight * <

C G @ 2eight * <

@ G 6 2eight * <

D G @ 2eight * B

A G C 2eight * =

@ G $ 2eight * =+ ? F! Weight > 6

E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$

$

"

S-ip$

S-ip

4

4

E l T Of : - l’ Al i h S E"


36/51

James Tam

Example Tra5e Of :rus-al’s Algorithm! Stop0 E"gesA55epte" > o o"es = 8


A G D 2eight * <

C G @ 2eight * <

@ G 6 2eight * <

D G @ 2eight * B

A G C 2eight * =

@ G $ 2eight * =

+ ? F! Weight > 6

E ? $! Weight > 3

D ? F! Weight > 7

* ? E! Weight > 8

(

A

)

*

B

+

,

&

&

$

$

"

S-ip$

S-ip

4

4

&

$

"

4

%

6

E l T Of : - l’ Al ith


37/51

James Tam

Example Tra5e Of :rus-al’s Algorithm!The Final Tree

(

A

)

*

B

+

,

&

&

$

$$

4


38/51

James Tam

Topologi5al Sorting Of $raphs! @eal ife ExampleB

•Ksed to order information that

could be represented as nodes in adirected acyclic graph.

•If theres a path from a node n to

another node n


39/51

James Tam

Topologi5al Sorting Of $raphs! @eal ife Example

•Ksed to order information that could be represented in the form

of the nodes in a directed acyclic graph

(P( $"&

(P( $""

(P( $"%

(P( $6% (P( "$%

(P( ""& ,'* "&&

Many se0or

courses

Many se0or

courses

t T l i l S ti + t * D With


40/51

James Tam

ote! Topologi5al Sorting +annot *e Done With+;5li5al $raphs

•Eorry no vacancies today.,

(P( $"& Admission into the

)epartment&

Computer Science 231 12"3$3&5

Introduction to Computer Science I

Problem solving and programming in a

structured language. )ata representation

program control basic 7ile handling the use o7

simple data structures and their

implementation. Pointers. 8ecursion.Prerequisite: Admission into the )epartment

o7 (omputer cience

Admission requirements into the

epartment of Computer Science

A grade o7 (3 or higher in (P( $"&

& his e9ample is purely 7ictional that was created to illustrate the principles o7 graph theory and should not be taken as an o77icial

description o7 peruisite reuirements 7or the )epartment o7 (omputer cience


41/51

James Tam

Algorithm For A Topologi5al Sort

public )ist topogicalEort #$raph g&

'

int i+

int no%odes * g.get%umber%odes #&+

)ist ordered%odes * new )ist #&+

%ode temp+

for #i * 5+ i - no%odes+ i&

'

temp * g.get%eFtTopParent #&+

ordered%odes.add #temp&+

g.delete%ode6dges #temp&+

4

4


42/51

James Tam

Example Of A Topologi5al Sort

(

A

)

*

B

+

,

-

& &

" $

" $

Graph

!ist

@emo e .+/ An" E"ges From $raph An"


43/51

James Tam

@emoCe .+/ An" E"ges From $raph An"A"" To ist

(

A

)

*

B

+

,

-

& &

" $

" $

Graph

!ist

051 01 0


44/51


45/51

James Tam

1p"ate The umbers An" @emoCe .*/

)

*

B

+

,

-

& $

$ $

Graph

!ist

051 01 0


46/51

James Tam

1p"ate The umbers An" @emoCe .D/

)

*+

,

- &

$ $

Graph

!ist

051 01 0


47/51

James Tam

1p"ate The umbers An" @emoCe .E/

*+

,

-

& &

Graph

!ist

051 01 0


48/51


49/51


50/51

James Tam

ou Shoul" o :no

•2hat is a minimum spanning tree

•Two algorithms #Prims and !rus"als& for creating minimum

spanning trees

•2hat is a topological sort and the algorithm for this sort.


51/51

Documents

Graphs Spanning