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CSE 203: Data Structures

Lecture 5: Graphs

Started on 09 April 2016 1Course Teacher: ro!" Sohel#ah$an

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Graphs

A graphG = (V,E) consists of a set ofverticesV, and a set of edgesE, such thateach edge inE is a connection between apair of vertices inV.

The number of vertices is written |V|, and thenumber edges is written |E|.


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Graphs (2)


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Paths and Cycles

Path: A sequence of verticesv1,v2, …,vn of length

n-1 with an edge fromvi tov

i+1 for 1

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Started on 09 April 2016 Course Teacher: ro!" Sohel#ah$an

5

• There is a si$ple path !ro$ &erte' 0 to &erte' 3 containin(

&ertices 0) 1) and 3"• &ertices 0) %) 2) 3) 1) %) 2) 3) 1 also !or$ a path) *ut not a

si$ple path *ecause so$e +ertices appear t,ice"

• &ertices 0) 1) %) and 0 !or$ a si$ple c-cle"

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Connected Components•

A su*(raph S is !or$ed !ro$ a (raph G = (V,E) *- selectin( a su*set V_s o! & a su*set E.so! E such that !or e+er- ed(e e in E.s) *oth o!e/s +ertices are in &.s"

• An undirected graph is connected if there is at leastone path from any vertex to any other.

• The maximum connected subgraphs of anundirected graph are called connected components.

Started on 09 Ap

ril 2016 6Course Teacher: ro!" Sohel#ah$an

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Ac-clic (raphs

• A (raph ,ithout c-cles is calledac-clic"

• Thus) a directed (raph ,ithout c-clesis called a directed ac-clic (raph orDAG"

• A !ree tree is a connected) undirected(raph ,ith no si$ple c-cles" – An eui+alent denition is that a !ree

tree is connected and has |V|-1 ed(es"


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Directed Representation


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Undirected Representation


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Co$parison


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Adjacency List Adjacency MatrixStores in!or$ation onl- !orthose ed(es that actuall-appear in the (raph

#euires space !or eachpotential ed(e) ,hether ite'ists or not

4eeds pointers" #euires no o+erhead !orpointers) ,hich can *e asu*stantial cost) especiall- i!the onl- in!or$ation stored!or an ed(e is one *it to

indicate its e'istence"Sparse (raphs are liel- toha+e their adacenc- listrepresentation *e $orespace e7cient"

8o,e+er) As the (raph*eco$es denser) theadacenc- $atri' *eco$esrelati+el- $ore space

e7cient"

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Space and Time

Adjacency Matrix: O(|V̂2|)

Adjacency List: O(|V| + |E|)


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Graph ADT

• e assu$e that – +ertices are dened *- an inte(er inde'

+alue"• there is a &erte' 0) &erte' 1) and so on

– There is a use!ul arra- called ar

C;;3elatest3"pd!


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atri'

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List

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Graph Traversals

Some applications require visiting everyvertex in the graph exactly once.

The application may require that vertices bevisited in some special order based ongraph topology.

Examples: –Artificial Intelligence Search –Shortest paths problems


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Graph Traversals Difficulties

• >irst) it $a- not *e possi*le to reachall +ertices !ro$ the start +erte'" – This occurs ,hen the (raph is not

connected

• Second) the (raph $a- containc-cles) and ,e $ust $ae sure that

c-cles do not cause the al(orith$ to(o into an innite loop"


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Graph Traversals Difficulties: Solutions

• Graph tra+ersal al(orith$s can sol+e *oth o! thesepro*le$s *- $aintainin( a $ar *it !or each +erte'on the (raph" – At the *e(innin( o! the al(orith$) the $ar *it !or all

+ertices is cleared"

– The $ar *it !or a +erte' is set ,hen the +erte' is rst+isited durin( the tra+ersal"

–

1

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Generic Graph Traversals Code


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Depth First Search (1)


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Depth First Search (2)


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D>S Cost

• D>S processes each ed(e once in adirected (raph"

• Sprocesses each ed(e !ro$ *othdirections"

• Each +erte' $ust *e +isited) *ut onl-

once"

• So the total cost is: Cost:Θ(|V| + |E|).


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S rocess


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C;;3elatest395"pd!

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Breadth First Search (1)

Like DFS, but replace stack with a queue. –Visit vertex’s neighbors before continuingdeeper in the tree.


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Started on 09 April 2016 2%Course Teacher: ro!" Sohel#ah$an

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S rocess


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C;;3elatest39"pd!

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Topological Sort (1)

Problem: Given a set of jobs, courses, etc.,with prerequisite constraints, output the

jobs in an order that does not violate any

of the prerequisites.


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• e can $odel the pro*le$ usin( aDAG" – The (raph is directed *ecause one tas is

a prereuisite o! another the +erticesha+e a directed relationship"

–

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• The process o! la-in( out the +ertices o! aDAG in a linear order to $eet theprereuisite rules is called a topolo(ical sort"

An accepta*le topolo(ical sort !or this

e'a$ple is: 1) 2) 3) %) 5) 6) "Started on 09 April 2016 29Course Teacher: ro!" Sohel#ah$an

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• A topolo(ical sort $a- *e !ound *-per!or$in( a D>S on the (raph"

• hen a +erte' is +isited) no action is taen

i"e") !unction PreVisit does nothing).• When the recursion pops *ac to that

+erte') !unction PostVisit prints thevertex.

• This -ields a topolo(ical sort in re+erseorder"


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DFS Based Topological Sort


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Course Teacher: ro!" Sohel#ah$an 32

Topolo(ical Sortin(

E'a$ple

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Topolo(ical Sortin(

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Course Teacher: ro!" Sohel#ah$an 3%

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Topolo(ical Sortin(

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Topolo(ical Sortin(

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Topolo(ical Sortin(

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Course Teacher: ro!" Sohel#ah$an %1

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Queue-Based Topsort• >irst +isit all ed(es) countin( the nu$*er o! ed(es that

lead to each +erte' – i"e") count the nu$*er o! prereuisites !or each +erte'

• All +ertices ,ith no prereuisites are placed on theueue" And ,e *e(in processin( the ueue"

• hen &erte' & is taen o o! the ueue) it is printed)and all nei(h*ors o! & ha+e their counts decre$ented *-one" – All nei(h*ours ha+e & as a prereuisite

•

lace on the ueue an- nei(h*or ,hose count *eco$esFero"

•

– i"e") there is no possi*le orderin( !or the tass that

does not +iolate so$e prereuisiteStarted on 09 April 2016 %2Course Teacher: ro!" Sohel#ah$an

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Queue-Based Topsort

cse 203-lec7

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