Unit 6 Finalbfg

7/21/2019 Unit 6 Finalbfg

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(1)32 51 27 85 66 23 13 57

(2)32 51 27 85 66 23 53 57 (EXCHANGE 27<85)

(3)32 27 51 85 66 23 13 57

(4)32 27 51 85 66 23 13 67 (EXCHANGE 66<85)

(5)32 27 51 66 85 23 13 57 (EXCHANGE 23<85)

(6) 32 27 51 66 23 85 13 67 (EXCHANGE 13<85)

(7) 32 27 51 66 23 13 85 57 (EXCHANGE 13<85)

(8) 32 27 51 66 23 13 57 85

S"!"CTI#( S#$T

5 selection sort is one in *hich successi/e ele!ent are selected in order and laced into

their roer osition The selection sort is used to find the s!allest ele!ent along a "#$% a"l-#$%''a"n$% and interchange it *ith 5"#$ The interchange laces the s!allest ele!ent

is the first osition in the table Si!ilarl again locate the second s!allest ele!ent Then

this ele!ent is laced at 5"89#$

The rocess of location or finding for the s!allest ele!ent continues k until all the

ele!ents ha/e been sorted in ascending order if n ele!ent are resent then (:-#)

co!arison is required Hence the nu!ber of co!arison are roortional to n& ie ;(n&)

EXAMPLE

PASS 1: 11 56 47 36 24 91 85 32

PASS 2: 11 24 47 36 56 91 85 32

PASS 3: 11 24 32 36 56 91 85 47

PASS 4: 11 24 32 36 47 91 85 56

PASS 5: 11 24 32 36 47 56 85 91

PASS 6: 11 24 32 36 47 56 85 91

PASS 7: 11 24 32 36 47 56 85 91

+!,#$IT:

S<84S.RT (5% :% MI:% T<M=)

ST"% 1: ST5RT

ST"% &: R<=<5T< ST<= >% ? @.R IA# T. : B #

ST"% /: S<T MI:A5"i$

ST"% 0: R<=<5T< ST<= @.R CAi9# T. : B #

I@ 5 "C$ ,MI:% TH<:

S<T T<M=A 5"D$

5"D$ A MI:



MI: A T<M=

EAD

<:0 .@ I@

<:0 .@ ST<= > 8..=

ST"% : I@ 5"i$ ,MI:% TH<:

S<T T<M=A5"i$

5"i$ AMI:

5"E$ AT<M=

"<:0 .@ I@$

"<:0 .@ S<= & 8..=S$

ST"% 6: <EIT

2IC3 S#$T:

Quick sort is a sorting algorith! *hich is based on di/ide and conquerrule The three ste

di/ide and conquer is gi/en belo* to sort tical sub arra5 "r$

4I5I4":

The arra 5"='R$IS =ortioned (rearranged) into t*o non-e!t sub arra 5"=Q$ anda "Q9#'R$ such that each ele!ent of 5 "'q$ is less than or equal to each ele!ent of 5

"q9#'r$ The index q is co!uted as art of this ortioning rocedure

Conquer:

The t*o sub arras 5"'q$ and 5"q9#'r$ are sorted b recursi/e calls to quick sort

Comine:

Since the sub arras are sorted in lace% no *ork is needed to co!bine the!% the entire

arra a "r$ Is no* sorted

The follo*ing function i!le!ents the quick sort

2IC3 S#$T '+7p7r):

Step 1: if (,r) then

QA ortioned (5%%r)

Step&: quick sort (5%%q)

Step /:Quick sort (5%q9#%r)

Step 0: sto:

%ortioned list:

:o* *e !ust construct the function ortioned There are se/eral !ethod that *e !ight

us % !ethod that so!eti!es are faster than the algorith! *e de/elo but that are intricate

and difficult to get correct The algorith! *e de/elo is !uch si!ler and easier to

understand% and it is certainl not slo*F in fact it does the s!allest ossible nu!ber of

ke co!arisons of an ortioning algorith!

Choice of pivot:



3e are not bounded to the choice of the first ite! in the list as the i/otF *e can choose

an ite! *e *ish a s*a it 3ith the first ite! before beginning the loo that ortions the

list In fact first ite! is often a oor choice for i/ot Since if the list is alread sorted%

then the first% ke *ill ha/e no other less than it% and so one of the sub list *ill be e!t

Hence let us instead choose a i/ot near the center of the list% I the hoe that our artition

the kes so that about ha/e co!e on each side of i/ot.

%+$TI#(I(, +$$+8G

The ke to algorith! is =5RTI.:<0 function% *hich rearrange the sub arra

5"r $ in the lace

%+$TITI#( '+7%7$):

Step 19: xA5"$%iA-#%DAr9#

Step &: as long as 5"D$,Ax%DAD-#%

Step /: incre!ent 6i7 /alue as long as

5"i$Ax

Step 0: if (i,D) then

S*a 5"i$ and 5"D$

<lse return D

Step : if (i,D) then sto ste &

Step 6: sto

I(S"$TI#( S#$T:Suose arra 5 *ith n ele!ents 5 "#$%5 "&$'''5":$ the insertion sort algorith!

scans 5 fro! 5"#$ T. 5":$%inserting each ele!ent 5 "$ into its roert osition is the

re/iousl sorted arra "#$%5"&$'''5"-#$ That is%

%+SS 1: 5"#$ b itself is tri/iall sorted

%+SS &: 5"&$ is inserted either before or after 5"#$ so that 5"#$%5"&$ is sorted

%+SS /: 5"$ is inserted into its roer osition is 5"#$%5"&$ that is before 5"#$%bet*een

5"#$ and 5"&$ or after 5"&$%so that 5"#$%5"&$ and 5"$ is sorted

%+SS 0: 5">$ is inserted into its roer osition in 5"#$%5"&$%5"$%5">$ is sorted

-

--

--

%+SS (: 5":$ is inserted into its aer lace in 5"#$%5"&$%''''5":-#$ so that

5"#$%5"&$%'''5":$ is sorted

The idea of this !ethod is to be laced an unsorted ele!ent into its correct osition in

a gro*ing sorted list of data



S.RT<0 8IST J:S.RT<0 8IST

+!,#$IT:

# 5ccet n nu!ber into arra data

& & 0ata ";$ is considered a sorted file of one ele!ent

:extA#

> :e* ele!ent A data"next$

? Mo/e all ele!ent K ne* ele!ent b one osition to the rightL Insert ne* ele!ent in the arra at the osition *here ste ? is ter!inated

K :ext next9#

N +ontinue fro! ste > as long as next ,n

O sto

erge sort:

It basicall *orks on the rincile of di/ide and conquers technique In the !ethod

first *e di/ide the list in t*o sub list% *ith each sub list of al!ost equal siPes +ontinue

di/iding the list till sub list reduce to unit length Merge the sub list *hich *ill be in

sorted order

"*ample :

Sort the follo*ing set of nu!ber% using !erge sort !ethod

>; L; K; ?; &; #; ;

3e no* break the list in t*o sub lists

@irst sub listG >; L; K; ?;

Second sub listG&; #; ;

5gain subdi/ide first sub listG >;% L; and K;%?;

@urther di/ision gi/es us

, and ,

And

40 60 70 50

40, 60 50,7



Me!e "#e$e "%&:

N&% d''de "#e $e&nd $*+'$

And 30

And

Merge

Merge

:o* !erge the t*o sub listG

$adi* Sort:

In this sorting records or ele!ents of the table are arranged in sequential order

@or nu!eric data of deci!al nu!ber % suose there are O ackets corresonding to the

deci!al digits are to be sorted

>N%#>%L#%>&%?N%#&N%&#%?>%LL

These nu!ber *ould be sorted in three hase

# In first ass% the in it digits are sorted into ackets The cards are collected

ackets % fro! O to acket ;& In second ass% the #; digits are sorted into ackets

In the third and final ass% the #;; digits are sorted into ackets

Chapter-1; ,raphs

Introduction:

40, 50,

20, 10

20 10 30

10, 20 30

10, 20, 30

40, 50, 60, 10, 20, 30

10 20 30 40 50 60



In this chater *e *ill be another non-linear data structureG the grah *hich is *idel

used to sol/e !an real life roble!s Basicall the grah is collection nodes called

/ertices and line seg!ents connecting airs of /ertices called line or edges

4efinition and Terminology:

,raph:

4efinition:

• 5 grah is a collection of t*o sets and <% *here

a 5 set of ele!ents called nodes (or oints or /ertices)

b Set < of edges such that e in < is identified *ith unique (unordered) air "u%/$

of nodes in % denoted b eA"u%/$

• 5 grah can be denoted b A (% <) or an edge is reresented b t*o adDacent

/ertices is reresented as A "% <$

• In abo/e definition e reresents edge u and / are endoints in e % u and / are said

to be adDacent or neighbours

"*ample1:

,< '57 ")

5< =+7 B7 C7 4>

"< ?'+7 B)7'B7C)7'C74)7'47+)@

Aig: .1 "*ample of graphHere 5% B% +% 0 are nodes% lines bet*een 5% B% +% 0 are called as edges

.&.& Types of graphs:

There are t*o tes of grahs

(a) Jndirected grah b 0irected grah

a. ndirected graph :

5n undirected grah is a grah in *hich there is no direction on the lines

%kno*n edgesIn an undirected grah % the flo* bet*een t*o /ertices can go in

either direction

ndirected graph

5n undirected grah (reresented as A(%<)) is one *here each edge < is

an unordered air of /erticesThus the air

(#%&) and (&%#) reresent the

sa!e edge

"*ample &:

A

D

C

B



Aig :.& :

The abo/e grah # and & are undirected grah *here & is also a tree Tree are secial

case of grahs

@or #

#A(%<) (#)A5%B%+%0

<(#)A(5%B)%(5%+)%(5%0)%(B%+)%(B%0)%(+%0)

@or &

&A(%<)

(&)A5%B%+%0%<%@%

<(#)A(5%B)%(5%+)%(5%0)%(B%<)%(+%@)%(+%)

') 4irected graph :

• 5 directed grah is also called as diagrah or grah 5 directed

grah is one *here each edge is reresented b a secific

direction or b a directed air or each edge is reresented b a

air of ordered /ertices ie an edge has a secific direction

• Suose is directed grah *ith directed edges eA(u%/)Then e

is also called as an arc@ollo*ing ter!inolog used G

(i) e begins at u and ends at /

(ii) u is the origin or initial oints of e % and / is the destination or ter!inal oint of e

(iii) u is a redecessor of / and / is successor or neighbour at u

(i/) u is a adDacent to / and / is adDacent to u

(/) The out degree of a node is the nu!ber of edges beginning at u(/i) The indegree of a node is the nu!ber of edges ending at u

(/ii) 5 node is called a source if it is has a ositi/e outdegree but Pero indegree

(/iii) 5 node is called sink if it has a Pero outdegree but ositi/e indegree

A(%<)

A"5%B%+%0$



< Ae#%e&%e%e>%e?%eL

@ig G N 0irected grah Ter!inolog G

'a) 4egree :

It is the nu!ber of edges containg a node It is *ritten as deg(u)% *here u is a

nodeIf deg(u) A;%

Then u does not belong to an edge Then u is called an isolated node

') Isolated verte* :

5s said abo/e % if an node does not bellog too an edge then it is called as

isolated node

<xa!le G :ode < is isolated grah



@ig G N> rah

'c) %ath :

It is sequence of /ertices in *hich each /ertex is adDacent /ertices to the next

one

'd) !ength of path :

The length of a ath is the nu!ber of edges on it'e) !inear path :

5 linear ath is a ath *hose first and last /ertex are distinct'f) Cycle:

5 ccle is a closed grah *ith length or !ore % such that starts and ends *ith

sa!e /ertex

@ig G rah *ith ccle

'g) Complete graph :

5 grah is said to be co!lete if e/er node in grah is adDacent to e/er

other node 5 co!lete grah *ith n nodes *ill ha/e "n(n-#)U&$ edges'h) Connected graph:

T*o /ertices /i and / D are said to be connected if there is a ath in fro! / i and

/ D

'i) Sugraph :

5 subgrah of is a grah 2 such that (2) () and <(2) <()

"*ample0:

A

A C



Aig :'+ ),raph ,1 'B) Sugraph of ,1

'D) !aeled graph :

5 grah is said to labeled if it2s edges are assigned data

'E) Feighted graph :

5 grah is said to be *eighted if each edge e in is assigned a non-negati/e

nu!erical /alue 3(e) called *eight or length of e in such a case % each ath =

in is assigned a *eight or length *hich is the su! of the *eights of the

edges along the ath =

'l) ultiple edge :

If one or !ore edges connects to the sa!e endoints then edges

e&

1G e/e1

0

@ig GNK G 3eighted grah @ig G Multile edges

Reresentation of grah G

There ar t*o standard *as of !aintaining a grah in the !e!or of

co!uter# Sequential reresentation of % is b !eans of its adDacenc !atrix of 5

& 8inked reresentation of % is b !eans of linked list of neighbours

Sequential reresentation of rah GSequential reresentation of grah is done b adDacenc !atrix

(a) 5dDacenc !atrix G

• The adDacenc !atrix 5 of grah is a t*o di!ensional

arra nVn ele!ents *here n is the nu!ber of /ertices Then

entries of 5 are defined as %

5"i$"D$A# %if there exists an edge bet*een /ertex I and D in 5"i$"D$A; %if no edge exists bet*een I and D

C .

A

.C

C

A



• Suose is a si!le directed grah *ith ! nodes and

suose nodes of ha/e been ordered and are called

#%&'''' !Then the adDacenc !atrix 5A(aiD) of the

grah is the nVn !atrix defined as follo*s

5iD A # if /i is adDacent to /D % that I % if

there is an edge (i % D)

; other*ise

Such a !atrix % *hich contains entries of onl ;

and # % is called a Bit !atrix or a Boolean !atrix

• The adDacenc !atrix 5 of grah does not deends on the

ordering of the nodes of % that is a different ordering of the

nodes !a result in a different adDacenc !atrix Ho*e/er %

the !atrices resulting fro! t*o different ordering are closel

related is that % one can be obtained fro! the b si!l

interchanging ro*s and colu!ns

.ther*ise % assu!e that the nodes of the grah ha/e fixed

ordering

• If grah is an undirected grah then adDacenc !atrix *ill

be a s!!etric !atrix

• @or exa!le :

Aig :graph

<xa!le L G

+onsider the grah gi/en

1 &

/

(#) i/e adDacenc !atrix reresentation

(&) i/e adDacenc list reresentation

A

C

.



#)5dDacenc !atrix A

&) 5dDacenc list G

Start

1 & /

& / 1

/ & 1

"*ample :

,iven adDacency matri*

E/ae: & nd "#e a"# a"'/ & !a# !'en +e&%

A

.C



.'e"ed !a#

A1A2A3A

A a+&e !a# G #a$ 4 n&de4

4A1A2A3A4

And ea'n! n&n-e& en"'e$ 'n 4 + 1

;e &+"a'ned "#e a"# a"'/ P & !a# G

A1

A2

A3

A4

0

1

1

0

0

0

0

0

0

1

0

1

1

1

1

0

0 0 1 0

1 0 1 2

0 0 1 1

1 0 0 1

1 0 0 1

1 0 2 2

1 0 1 1

0 0 1 1

0

2

1

2

0

0

0

0

0

2

1

1

1

3

2

1



4

ea'n! n&ne& eeen"$ + 1,

4

The abo/e grah is not strongl connectedF since the ath

Matrix = of has Pero entries

"*ample: 3rite 3arshall2s algorith! (*rite co!!ents *here/er alicable) for

grah sho*n belo*

')eae ad=aen a"'/

'') Peae a"# a"'/ a'n! ;a$#a>$

a!&'"#

1 0 2 3

5 0 6 8

3 0 3 5

2 0 3 5

1 0 1 1

1 0 1 1

1 0 1 1

1 0 1 1



')Ad=ane a"'/

?1 ?2 ?3 ?4

A

?2 ?3

?4

'') Peae a"# a"'/ a'n! ;a#a>$ a!&'"#

A1

A2A@A

A2

A3

/

0 1 0 1

0 0 1 0

0 0 0 0

0 1 1 0

0 1 0 1

0 0 1 0

0 0 0 0

0 1 1 0

0 1 0 1

0 0 1 00 0 0 0

0 1 1 0

0 1 0 1

0 0 1 0

0 0 0 0

0 1 1 0

X



A3

A4

4A1A2A3A4

4

Pa"# a"'/

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 2 3 1

0 0 1 0

0 0 0 0

0 1 0 0

0 1 1 1

0 0 1 0

0 0 0 0

0 1 1 0



4"%T AI$ST S"+$C:

The idea behind a deth first search beginning at a starting node 5 is as follo*sG first *e

exa!ine the starting node 5 then *e exa!ine each node : along ath = *hich begins at5% that is *e rocess a neighbor of 5% the neighbor of neighbor of 5 and so on

5fter co!ing to a 6dead end7 that is to the end of ath = % *e backtrack on = until *e can

continue along another ath = and so on

This algorith! is si!ilar to breadth first search excet *e uses stack% instead of queue

5gain% a field ST5TJS is used to tell us the current status of a node

+lgorithm:

This algorith! executes a deth first search on a grah beginning at the startingnode 5

Step 1: initialiPe all nodes to read state (ST5TJS A#)

Step &: ush the starting node 5 into ST5+ and change its status to the *aiting state

(ST5TJS A&)

Step /: reeat ste > and ? until stack is e!t

Step 0: o the to node : of stack =rocess : and change its status to the rocessed

state (ST5T< A)

step : ush onto stack all the neighbors of : that are still in the read state (ST5TJS

A#)%and change their status to the *aiting state (ST5TJS A&)

"<nd of ste loo$

Step 6: <xit

EXAMPLE: C

A

E

.

N&de Ad=aen '$"

A C,

-

C

. A,,C,E



E C

E/ana"'&n :

n'"'a *$# n&de &n "#e $"aB

P& and 'n" "#e "& eeen" and "#en *$# &n"& "#e $"aB a "#e

ne'!#+&$ & 'n ead $"a"e

*" 'n a$e n&de d&e$n>" #ae an Ne'!#+&$ and $& n&"#'n! '$ *$#ed &n"& $"aB

and e" $"aB '$ e" S& "#e a!&'"# &e"e$

#ee&e "#ee '$ n& n&de ea#a+e & n&de

EXAMPE 23:

C&n$'de "#e !a# G !'en $*&$e %e %an" "& nd and 'n" a "#e n&de$

ea#a+e & "#e n&de =('n*d'n! D '"$e)

ne %a "& d& "#'$ '$ "& *$e a de"# $" $ea# & G $"a"'n! a" n&de D "#e $"e ae

a$ &&%$,

a) n'"'a *$# D &n"& $"aB

&

$"aB

+) P& and 'n" "#e "& eeen" D and "#en *$# &n"& "#e $"aB a "#e

ne'!#+&$ & D("#&$e ae 'n "#e ead $"a"e) a$ &&%$,

(C) P& and 'n" "#e "& eeen" B, and "#en *$# &n"& "#e

S"aB a "#e ne'!#+& & B 'n ead $"a"e

D

F

.

G

E

.



PN P: D, F

(d) P& and 'n" "#e "& eeen" ! and "#en *$# &n"& "#e $"aB a "#e ne'!#+&$

& ! 'n ead $"a"e,

C

E

.

(e) PP AN. PN P EEMEN C AN. HEN PSH N HE SACF A

HE NEGHS C N EA.I SAE

() P& and 'n" "& eeen" and "#en *$# &n"& "#e $"aB a "#e ne'!#+&$> &

'n ead $"a"e

E

.

E

.



PN P: D, F, G, C,

N&"e "#a" "#e &n ne'!#+& . & '$ n&" *$#ed &n"& "#e $"aB, $'ne . '$ &n 'n

ead $"a"e

(!) & and 'n" "& eeen" E and "#en *$# &n"& "#e $"aB a "#e ne'!#+&*$ &

E Een E #a$ 3 ne'!#+&*$ +*" n&" 'n ead $"a"e

P'n" &*"*": D, F, G, C , E

(#)& and 'n" "#e "& eeen" . and *$# a" $"aB a ne'!#+&*$ & . $'ne .

ne'!#+&*$ ae n&" 'n ead $"a"e and $"aB '$ e", $& "#e de"# $" $ea# &

G $"a"'n! a" D '$ n&% &e"ed

D, F, G, C, , E, . ae n&de$ ea#a+e & n&de D

EXAMPLE:

'nd and 'n" a n&de$ ea#a+e & n&de F & &&%'n! !a#:

(H'n" *$e de"# $" Sea# S#&% &n"en" & $"aB a" "#e end & ea# $"e

%'"# +'e E/ana"'&n)

.



C&n$'de "#e a+&e !a# and nd a "#e n&de$ ea#a+e & F

a n'"'a *$# F &n"& "#e S"aB

SACF

+ P& and 'n" "#e "& eeen" F, and "#e *$# &n"& "#e $"aB a "#ene'!#+&*$ & F 'n ead $"a"e

P& and 'n" "#e "& eeen" G, and "#en *$# &n"& "#e $"aB a

"#e ne'!#+&*$ & ! 'n ead $"a"e

P'n" &*"*" : F,G

d P& and 'n" "#e "& eeen" C and "#en *$# a "#e ne'!#+&*$ & C 'n

ead S"a"e

P'n" &*"*": F, G, C

e P& and 'n" "#e "& eeen" and "#en *$# a "#e ne'!#+&* & 'n ead

$"a"e

P& and 'n" "#e "& eeen" E and *$# &n"& "#e $"aB a "#e

ne'!#+&* & E een E #a$ 3 ne'!#+&* +*" n&" 'n ead $"a"e

! P& and 'n" "#e "& eeen" and *$# &n"& "#e $"aB a "#e

ne'!#+&* & $'ne ne'!#+&* ae n&" 'n ead $"a"e and $"aB '$ e", $&

"#e de"# $" $ea# & !a# $"a"'n! a" F '$ n&% &e"e

F

G

C

E



F, G, C, E, ae "#e n&de$ ea#a+e & n&de D

Applic!i"# "$ %rp&:

Ga# an +e *$ed 'n a'&*$ a'a"'&n 'Be

(a) ee$en"a"'&n & ee"' *en" J&%(+) Ma$ 'nd'a"e$ &nne"''" +e"%een d'Keen" ae$() ee#&ne and &*"e ne"%&B'n!(d) &*"'n! & &ne &a"'&n "& an&"#e(e) S#ed*'n! & 'n"edeenden" "a$B$ & a"''"'e$() C&*"'n! &=e"$ &e"'&n "'e, dea, ea, $"a" and a"e n'$#

"'e$ & a &=e" %#'# '$ ade * & $eea "a$B$(!) & nd $#&"e$" a"#, & "&&&!'a $&", '"'a a"# e"#&d

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Unit 6 Finalbfg