Chapter 9 data structure slide

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GraphsData Structures

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Formal definition of graphs

A graph G is defined as follows:

G=(V,E)

V(G): a finite nonempt! set of verticesE(G): a set of edges (pairs of vertices)

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Directed vs" undirected

graphs When the edges in a graph have no direction the graph is called

undirected

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Directed vs" undirected

graphs (cont") When the edges in a graph have a direction the graph is called

directed (or digraph)

E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7)

Warning : if the graph is

directed, the order of the

vertices in each edge is

important !!

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Trees vs graphs

Trees are special cases of graphs##

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$raph terminolog!

Ad%acent nodes: two nodes are ad%acent if the! are connected b! an

edge

&ath: a se'uence of vertices that connect two nodes in a graph omplete graph: a graph in which ever! verte is directl! connected

to ever! other verte

5 is adjacen ! 77 is adjacen "r!# 5

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$raph terminolog! (cont")

What is the number of edges in acomplete directed graph with * vertices?

N * (N-1)

2( )O N

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$raph terminolog! (cont")

Weighted graph: a graph in which each edge

carries a value

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$raph implementation

Arra!,based implementation

A -D arra! is used to represent the vertices

A .D arra! (ad%acenc! matri) is used to represent the edges

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Arra!,based implementation

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$raph implementation (cont")

/in0ed,list implementation

A -D arra! is used to represent the vertices

A list is used for each verte v which contains

the vertices which are ad%acent from v(ad%acenc! list)

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/in0ed,list implementation

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Ad%acenc! matri vs"ad%acenc! list representation

Adjacency matrix

onnectivit! between two vertices can be tested 'uic0l!

Adjacency list

1ertices ad%acent to another verte can be found 'uic0l!

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$raph specification based onad%acenc! matri representationc!ns in &'EDGE = %*

e#p+aec+ass -ere/0pec+ass Graph/0pe { pb+ic

Graph/0pe(in)* Graph/0pe()* !id aeE#p0()* b!!+ 8sE#p0() c!ns* b!!+ 8sF++() c!ns* !id dd-ere(-ere/0pe)*

!id ddEd:e(-ere/0pe, -ere/0pe, in)* in ;ei:h8s(-ere/0pe, -ere/0pe)* !id Ge/!-erices(-ere/0pe, +earars()* !id ar-ere(-ere/0pe)* b!!+ 8sared(-ere/0pe) c!ns*

priae in n#-erices* in #a-erices* -ere/0pe? erices*

in ??ed:es* b!!+? #ars*@*

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e#p+aec+ass -ere/0pe

Graph/0pe-ere/0peGraph/0pe(in #a-)

{

n#-erices = %* #a-erices = #a-*

erices = neA -ere/0peB#a-C*

ed:es = neA inB#a-C*

"!r(in i = %* i #a-* i)

ed:esBiC = neA inB#a-C*

#ars = neA b!!+B#a-C*@


Graph/0pe-ere/0peGraph/0pe()

{

de+ee BC erices*

"!r(in i = %* i #a-erices* i) de+ee BC ed:esBiC*

de+ee BC ed:es*

de+ee BC #ars*

@



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!id Graph/0pe-ere/0pe dd-ere(-ere/0pe ere){

ericesBn#-ericesC = ere*

"!r(in inde = %* inde n#-erices* inde) {

ed:esBn#-ericesCBindeC = &'EDGE*

ed:esBindeCBn#-ericesC = &'EDGE* @

n#-erices*@


!id Graph/0pe-ere/0pe ddEd:e(-ere/0pe "r!#-ere,

-ere/0pe !-ere, in Aei:h){

in r!A*

in c!+#n*

r!A = 8nde8s(erices, "r!#-ere)*

c!+ = 8nde8s(erices, !-ere)*

ed:esBr!ACBc!+C = Aei:h*@



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in Graph/0pe-ere/0pe;ei:h8s(-ere/0pe "r!#-ere,-ere/0pe !-ere)

{

in r!A*

in c!+#n*

r!A = 8nde8s(erices, "r!#-ere)*

c!+ = 8nde8s(erices, !-ere)*

rern ed:esBr!ACBc!+C*

@

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$raphs as ADTs

ADT graph operations

Test whether graph is empt!"

$et number of vertices in a graph"

$et number of edges in a graph" See whether edge eists between two given vertices"

2nsert verte in graph whose vertices have distinct values that differ from new

verte3s value"

Data Structures and Problem Solving with C++: Walls and Mirrors, >arran! and enr0, 2%13

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$raphs as ADTs

ADT graph operations ctd"

2nsert edge between two given vertices in graph"

4emove specified verte from graph and an! edges between the verte and other

vertices"

4emove edge between two vertices in graph"

4etrieve from graph verte that contains given value"

Data Structures and Problem Solving with C++: Walls and Mirrors, >arran! and enr0, 2%13

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$raph searching

Proble: find a path between two nodes of the graph (e"g" Austin

and Washington) !ethods: Depth,First,Search (DFS) or 5readth,First,Search (5FS)

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Depth,First,Search (DFS)

What is the idea behind DFS?

Travel as far as !ou can down a path

5ac0 up as little as possible when !ou reach a 6dead end6

(i"e" net verte has been 6mar0ed6 or there is no net

verte)

DFS can be implemented efficientl! using a

stac"

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Depth,First,Search (DFS)

(cont#)e "!nd ! "a+sesac.Hsh(sar-ere)DI sac.H!p(ere)

8F ere == end-ere e "!nd ! re EE Hsh a++ adjacen erices !n! sac

;8E Jsac.8sE#p0() &D J"!nd 8F(J"!nd) ;rie KHah d!es n! eisK

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sar end

(iniia+iLai!n)

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e#p+ae c+ass 8e#/0pe

!id DephFirsearch(Graph/0pe-ere/0pe :raph, -ere/0pe sar-ere,-ere/0pe end-ere)

{

ac/0pe-ere/0pe sac*


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e+se {

i"(J:raph.8sared(ere)) {

:raph.ar-ere(ere)* :raph.Ge/!-erices(ere, ere


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!id Graph/0pe-ere/0peGe/!-erices(-ere/0pe ere,


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5readth,First,Searching (5FS)

What is the idea behind 5FS?

/oo0 at all possible paths at the same depth before !ou go at a deeper

level

5ac0 up as $ar as possible when !ou reach a 6dead end6 (i"e" net verte

has been 6mar0ed6 or there is no net verte)

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5readth,First,Searching (5FS) (cont")

5FS can be implemented efficientl! using a%ueue

e "!nd ! "a+se

Mee.EnMee(sar-ere)

DI Mee.DeMee(ere)

8F ere == end-ere

e "!nd ! re

EE

EnMee a++ adjacen erices !n! Mee

;8E JMee.8sE#p0() &D J"!nd

Should we mar0 a verte when it is en'ueued orwhen it is de'ueued ?

8F(J"!nd) ;rie KHah d!es n! eisK

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sar end

(iniia+iLai!n)

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ne

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!id NreadhFirsearch(Graph/0pe-ere/0pe :raph, -ere/0pesar-ere, -ere/0pe end-ere)*

{


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e+se {

i"(J:raph.8sared(ere)) {

:raph.ar-ere(ere)*

:raph.Ge/!-erices(ere, ere


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Single,source shortest,path problem

There are multiple paths from a source verte to a destination verte

&hortest path: the path whose total weight (i"e" sum of edge weights)is minimum

7amples:

Austin,89ouston,8Atlanta,8Washington: -;< miles

Austin,8Dallas,8Denver,8Atlanta,8Washington: .=>< miles

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Single,source shortest,path problem(cont") ommon algorithms: 'i"stras algorithm ellan-+ord algorithm

5FS can be used to solve the shortest graph problem when the graphis weightlessweightless or all the weights are the same

(mar0 vertices before 7n'ueue)

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Chapter 9 data structure slide

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