15.spanning tree

7/25/2019 15.Spanning Tree

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Spanning Tree


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What is A Spanning Tree?

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A spanningtree for anundirected graph G=(V,E)is a subgraphof G that isa treeand contains all theverticesof G

an a graph have !orethan one spanning tree?

an an unconnected graphhave a spanning tree?


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"ini!al Spanning Tree#

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Mst T: w( T)= (u,v) Tw(u,v) is minimized

The $eightof a subgraphis the su! of the $eightsof it edges#

A !ini!u! spanning treefor a $eighted graph is aspanning tree $ith!ini!u! $eight#

an a graph have !orethen one !ini!u!spanning tree?


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E%a!ple of a &roble! that Translates into a"ST

The Problem Several pins of an electronic circuit !ust be

connected using the least a!ount of $ire#

Modeling the Problem The graph is a co!plete, undirected graph

G= ( V, E ,W), $here Vis the set of pins, Eis

the set of all possible interconnections bet$eenthe pairs of pins and $(e) is the length of the$ire needed to connect the pair of vertices#

'ind a !ini!u! spanning tree#


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Greed hoice

We $ill sho$ t$o $as to build a !ini!u!spanning tree#

A "ST can be gro$n fro! the currentspanning tree b adding the nearest verte%and the edge connecting the nearest verte% tothe "ST# (&ri!s algorith!)

A "ST can be gro$n fro! a forest of spanningtrees b adding the s!allest edge connectingt$o spanning trees# (*rus+als algorith!)


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Notation

Treevertices- in the tree constructed so far .ontree vertices- rest of vertices

Prims Selection rule

Select the minimum weight edge between a tree-

nde and a nn-tree nde and add t the tree


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The Prim algorithm Main Idea

Select a vertex to be a tree-node

while(there are non-tree vertices) {ifthere is no edge connecting a tree node

with a non-tree node returnno spanning tree

select an edge o! minim"m weight between atree node and a non-tree node

add the selected edge and its new vertex tothe tree#

ret"rn tree


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&ri!s Algorith!

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&ri!s Algorith!

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&ri!s Algorith!

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&ri!s Algorith!

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&ri!s Algorith!

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&ri!s Algorith!

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&ri!s Algorith!

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!ini!u! spanning tree


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*rus+als Algorith!

$r"s%al&s 'lgorithm

6# Each verte% is in its o$n cluster

5# Ta+e the edge e$ith the s!allest $eight if econnects t$o vertices in different clusters, then eis added to the "ST and the t$o clusters, $hich are connected b e, are !erged into a

single cluster if econnects t$o vertices, $hich are alread

in the sa!e cluster, ignore it

4# ontinue until n6edges $ere selected


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*rus+als Algorith!

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*rus+als Algorith!

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*rus+als Algorith!

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*rus+als Algorith!

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*rus+als Algorith!

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*rus+als Algorith!

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ccle77


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*rus+als Algorith!

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*rus+als Algorith!

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*rus+als Algorith!

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!ini!u! spanning tree


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Graph Traversal

Traversing a graph !eans visiting all thevertices in the graph e%actl once#

/readth 'irst Search (/'S)

0epth 'irst Search (0'S)


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0'SSi!ilar to inorder traversal of a binar

search tree

Starting fro! a given node, this traversalvisits all the nodes up to the deepestlevel and so on#


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v6

v8v5 v4

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0'S

S * +, - + - +. - +/0 +1 - +20 +30 +4


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v6

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0'S

S * +, - + - +. - +/0 +1 - +20 +40 +3


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0'S TraversalVisit the verte% v

Visit all the vertices along the path $hich begins at v

Visit the verte% v5 then the verte% i!!ediate

ad:acent to v, let it be vx# ;f vxhas an i!!ediatead:acent v6then visit it and so on till there is adead end#

ead end- A verte% $hich does not have ani!!ediate ad:acent or its i!!ediate ad:acent hasbeen visited#


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After co!ing to an dead end $ebac+trac+ to v to see if it has ananother ad:acent verte% other than vxand then continue the sa!e fro! it else

fro! the ad:acent of the ad:acent($hich is not visited earlier) and so on#


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&ush the starting verte% into the STA*

While ST'7$ not empt6 do

& all ad:acent verte% of +onto STA*

End of ;'

End of WhileST


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*

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Ad:acenc @ist

A- ',,//- G,

- '0- E- 0,,'- 0G- ,E- 0,**- E,G


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0'S of G starting at

6B ;nitiall push onto STA*STA* -

V;S;T- C

5B & onto the STA* allneighbor of

STA*- 0, *V;S;T-


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4B & all neighbor of * ontoSTA*

STA*- 0,E,G

V;S;T- , *

3B & all neighbor of G ontoSTA*

STA*- 0,E, E, ,

V;S;T- , *, G


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1B & all neighbor of ontoSTA*

STA*- 0,E,E, '

V;S;T- , *, G,

2B &


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1B & all neighbor of 0 ontoSTA*

STA*- 0,E,E,

V;S;T- , *, G, , ',0

2B &


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1B &


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*

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Ad:acenc @ist

A- ',,//- G,

- '0- E- 0,,'- 0G- ,E- 0,**- E,G

, *, G, , ', 0, E


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/'S Traversal

An verte% in label i $ill be visited onl

after the visiting of all the vertices inits preceding level that is at level iD 6


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/'S Traversal6B Enter the starting verte% vin a ueue

85B While 8is not e!pt do

0elete an ite! fro! 8, sa "

;f "is not in +ISITstore "in+ISIT

Enter all ad:acent vertices of "

into 84B Stop


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v6

v8v5 v4

v3

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v2v1


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6B ;nsert the starting verte% V6in F

F = V6

V;S;T = C

5B 0elete an ite! fro! F, let it be u = V6

u is not in V;S;T# Store u in V;S;T

and its ad:acent ele!ent in F F = V5 , V4

V;S;T = V6


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4B 0elete an ite! fro! F, let it be u = V5u is not in V;S;T# Store u in V;S;T and

its ad:acent ele!ent in F

F = + , V4 , +15 +.

V;S;T = V6 , +



F = +4 , V3, V15 +15 +3

V;S;T = V6 , V55 +4


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its ad:acent ele!ent in F F = +1, V1, V3, V2, +2

V;S;T = V6 , V5, V4, +1

2B 0elete an ite! fro! F, let it be u =V1u is not in V;S;T# Store u in V;S;T and


F = +., V3, V2, V8 , V9V;S;T = V6 , V5, V4, V3, +.


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9B 0elete an ite! fro! F, let it be u =V3

u is in V;S;T#

F = +1, V2, V8 , V9

V;S;T = V6 , V5, V4, V3, V18B 0elete an ite! fro! F, let it be u =V2

u is not in V;S;T# Store u in V;S;Tand its ad:acent ele!ent in F

F = +3, V8 , V9

=


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B 0elete an ite! fro! F, let it be u =V8u is not in V;S;T# Store u in V;S;T and

its ad:acent ele!ent in F F = +2 , V9, V6

V;S;T = V6 , V5, V4, V3, V1, V2, +2

6HB 0elete an ite! fro! F, let it be u =V9u is not in V;S;T# Store u in V;S;T and


F = +/, V6V;S;T = V6 , V5, V4, V3, V1, V2, V8,

+/


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66B 0elete an ite! fro! F, let it be u =V6u is in V;S;T#

F = +,

V;S;T = V6 , V5, V4, V3, V1, V2,V8, V9

65B F is e!pt, Stop

F =

V;S;T = V6 , V5, V4, V3, V1, V2,V8, V9


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/'S

15.spanning tree

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