algorithms lab manual

7/27/2019 Algorithms Lab Manual

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ALGORITHMS LAB

MANUAL

(MC9229)

Info Institute of EngineeringKovilpalayam

Coimbatore 641107.


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Table of Contents

S.No. List of Experiments Page No.

1 ui!" #ort $

2 %inary #ear!& 11

$ %inary Tree Traversal 1'

4 (ars&all)s *lgorit&m 2$

+ ,i-"stra)s *lgorit&m 2'

6 rim)s *lgorit&m $1

7 Knapsa!" roblem ,ynami! rogramming $6

/ #ubset roblem %a!"tra!"ing 41

' Travelling #alesman roblem %ran!& an %oun

10 #trassen)s atri ultipli!ation


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$

Aim3 Implement t&e #orting using ui!" #&ort met&o.

Algorithm3

T&e steps are3

1. i!" an element !alle apivot from t&e list.2. 5eorer t&e list so t&at all elements it& values less t&an t&e pivot !ome before t&e pivot

&ile all elements it& values greater t&an t&e pivot !ome after it e8ual values !an go

eit&er ay9. *fter t&is partitioning t&e pivot is in its final position. T&is is !alle t&epartition operation.

$. 5e!ursively sort t&e sub:list of lesser elements an t&e sub:list of greater elements.

T&e base !ase of t&e re!ursion are lists of si;e ;ero or one &i!& never nee to be sorte.In

simple pseuo!oe t&e algorit&m mig&t be epresse as t&is3fun!tion 8ui!"sortarray9

var list less greater

if lengt&array9 < 1

return arraysele!t an remove a pivot valuepivot from array

for ea!& in arrayif < pivot t&en appen to less

else appen to greater

return !on!atenate8ui!"sortless9 pivot 8ui!"sortgreater99

Program3

=in!lue>stio.&>=in!lue>!onio.&>

int i-npivota?20@A

voi 8ui!"int a?@int leftint rig&t9Avoi sapint a?@int iint -9A

voi main9

Bint na?20@A

tet!olor1+9A

!lrs!r9A

printf>nnDICK #5T>9Aprintf>nnEnter t&e limit 3 >9A

s!anf>F>Gn9A

tet!olor49Atet!olor+9A

!lrs!r9A

printf>nnEnter t&e elementnn>9AforiH0Ai

s!anf>F>Ga?i@9A

8ui!"a0n:19A

tet!olor109A


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printf>nnT&e sorte list is nn>9A

foriH0Ai

printf>F >a?i@9Aget!&9A

voi 8ui!"int a?@int firstint last9B

iffirst

BpivotHa?first@A

iHfirstA

-HlastA

&ileiB

&ilea?i@JHpivotGGi

iA

&ilea?-@LHpivotGG-Lfirst9-::A

ifisapai-9A

sapafirst-9A8ui!"afirst-:19A

8ui!"a-1last9A

voi sapint a?@int iint -9

B

int tempAtempHa?i@A

a?i@Ha?-@A

a?-@HtempA

SAMPLE INPUT AND OUTPUT:

Enter t&e limit 3 +

Enter t&e elements+

4

$2

1

T&e sorteMlist is1 2 $ 4 +


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2. Aim3 To implement %inary #ear!&

Algorithm3BinarySearchA?0..n : 1@ key lo hi9

MM Input3 *n integer sorte arrayA?0..n 1@.

MM Output3 returns mid if "ey foun9 or :1 "ey not foun9if lo L hi

return :1 MM "ey not foun

mid N low high9M2

if key HA?mid@ return mid MM "ey foun

if key JA?mid@

returnBinarySearch A n key lo mid 19

else

returnBinarySearch A n key mid 1 hi9

enBinaySearch.

Program3

=in!lue>stio.&>=in!lue>!onio.&>

voi main9

Bint a?2+@i-tempsnlomi&ig&A

!lrs!r9A

printf>nEnter t&e Oimilt 3 >9A

s!anf>F>Gn9Aprintf>nnEnter t&e elementsn>9A

foriH0Ai

B

s!anf>F>Ga?i@9A

foriH0Ai

B

for-H0A-

Bifa?-@La?-1@9

B

tempHa?-@Aa?-@Ha?-1@A

a?-1@HtempA

printf>nn#orte list>9A

foriH0Ai


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B

printf>nF>a?i@9A

printf>nnEnter t&e elements to be sear!&e 3 >9A

s!anf>F>Gs9A

&ig&Hn:1AloH0A

&ileloJH&ig&9

BmiHlo&ig&9M2A

ifsLa?mi@9

loHmi1A

else ifs&ig&Hmi:1A

else ifsHHa?mi@9

B

printf>nnT&e element F is foun>s9Aget!&9A

eit09A

printf>nnT&e element F is not foun>s9Aget!&9A

SAMPLE INPUT AND OUTPUT:

Enter t&e elements+

4

$2

1

#orte list1

2

$

4+

Enter t&e element to be sear!&e 3 +

T&e element + is foun.


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$. Aim3

To rite a C:program to implement binary tree traversal.

Des!ription3

T&e most basi! traversal algorit&ms for general trees arepreordertraversal anpostorder

traversal is!usse in #e!tions 7.2.2 an 7.2.$ respe!tively of t&e Poori!& an Tamassia tet.Essentially a preorer traversal of a tree involves visiting a noe of t&e tree starting at its root

an t&en preorer traversing t&e subtrees roote at ea!& of its !&ilren. * postorer traversal on

t&e ot&er &an involves first postorer traversing t&e subtrees roote at ea!& of t&e !&ilren of anoe an t&en visiting t&e noe itself starting at t&e root. T&ese algorit&ms are given in

pseuo!oe belo. Qote t&at t&e !alls preorder(T,T.root())an postorder(T,T.root())

perform preorer an postorer traversals respe!tively of an entire tree T.

AlgorithmAlgorithmpreorder(T, v)

Input:A tree T and a node v of T.

Output:Depends on the action performed on a visit to a node.

visit node v

for each node w in T.children(v)

preorder(T,w) // recursivel traverse the su!tree rooted at the child w

of v

Algorithmpostorder(T, v)

Input:A tree T and a node v of T.

Output:Depends on the action performed on a visit to a node.

for each node w in T.children(v) postorder(T,w) // recursivel traverse the su!tree rooted at the child w

of v

visit node v

"oing:

=in!lueJstio.&L

=in!lueJ!onio.&L

=in!lueJstlib.&L

typeef stru!t treenoe

B

int ataA


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stru!t treenoe RleftA

stru!t treenoe Rrig&tA

tnoeA

tnoe RinsertioninttnoeR9A

voi preorertnoe R9A

voi inorertnoe R9A

voi postorertnoe R9A

voi main9

B

tnoe RTHQDOOA

int !&1nA

!&ar !&2A

o

B

!lrs!r9A

printf>nttRRRRperation (it& TreeRRRR>9A

printf>nt1.Insertion>9A

printf>nt2.Inorer Traversal>9A

printf>nt$.reorer Traversal>9A

printf>nt4.ostorer Traversal>9A

printf>ntEnter Sour C&oi!e 3>9A

s!anf>F>G!&19A

sit!&!&19


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B

!ase 13

printf>nn Enter t&e element to be inserte 3>9A

s!anf>F>Gn9A

THinsertionnT9A

printf>Q is F>n9A

brea"A

!ase 23

inorerT9A

brea"A

!ase $3

preorerT9A

brea"A

!ase 43

postorerT9A

brea"A

efault3

printf>nnInvali ption>9A

brea"A

printf>nn,o you ant to !ontinue yMn3>9A

s!anf>Fs>G!&29A

&ile!&2HHy9A


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get!&9A

tnoe Rinsertionint tnoe RT9

B

ifTHHQDOO9

B

THtnoe R9mallo!si;eoftnoe99A

ifTHHQDOO9

printf>nout of spa!e>9A

else

B

T:LataHA

T:LleftHT:Lrig&tHQDOOA

else

B

ifJT:Lata99

T:LleftHinsertionT:Lleft9A

else

B

ifLT:Lata9

T:Lrig&tHinsertionT:Lrig&t9A


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return TA

voi preorertnoe RT9

B

ifTUHQDOO9

B

printf>tF>T:Lata9A

preorerT:Lleft9A

preorerT:Lrig&t9A

voi postorertnoe RT9

B

ifTUHQDOO9

B

postorerT:Lleft9A

postorerT:Lrig&t9A

printf>tF>T:Lata9A

voi inorertnoe RT9


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B

ifTUHQDOO9

B

inorerT:Lleft9A

printf>tF>T:Lata9A

inorerT:Lrig&t9A

OUTPUT:

RRRRperation (it& TreeRRRR

1.Insertion

2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal

Enter Sour C&oi!e 31

Enter t&e element to be inserte 324

Q is 24

,o you ant to !ontinue yMn3y


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1.Insertion

2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal


Enter t&e element to be inserte 3$4

Q is $4



1.Insertion

2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal


Enter t&e element to be inserte 340

Q is 40



1.Insertion


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2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal


24 $4 40



1.Insertion

2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal

Enter Sour C&oi!e 3$

24 $4 40



1.Insertion

2.Inorer Traversal

$.reorer Traversal

4.ostorer Traversal


40 $4 24

,o you ant to !ontinue yMn3n


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4. Aim3

To implement (ars&all)s algorit&m

Des!ription3

T&e input to t&is algorit&m is an a-a!en!y matri &i!& &as 1s for any ro:!olumn pair

if t&ere eists an ege beteen t&em. (&at ell be oing &ere is :: !&e!"ing out for everyverte t&e in!oming an outgoing lin"s. If t&ere eist bot& in!oming an outgoing lin"s for a

parti!ular verte e ill put a 1 for t&e ro:!olumn pair forme by t&em in t&e a-a!en!y

matri.

* % C , E

* 0 0 0 1 0 *,9

% 0 0 0 1 0 %,9

C 0 1 0 0 0 C%9

, 0 0 0 0 1 ,E9E 0 0 1 0 0 EC9

,ire!te Prap& *-a!en!y atri

Oets ta"e t&e verte E into !onsieration. Sou &ave an ege !oming from , an you &ave anege going to CA 8uite visible from t&e matri. #o as I sai ell put a 1 for t&e element ,C in

t&e matri. #imilarly for t&e verte C ell put 1 for t&e element E%. (ell o t&is for all t&e

verte in t&e grap& an &at you get at t&e en is t&e Transitive Closure of t&is grap&. T&is ay

of fining t&e transitive !losure as erive by (ars&all an t&us t&e name (ars&allsalgorit&m.

Algorithm:

#tep:13 Copy t&e *-a!en!y matri into anot&er matri !alle t&e at& matri

#tep:23 Vin in t&e at& matri for every element in t&e Prap& t&e in!oming an outgoing eges#tep:$3 Vor every su!& pair of in!oming an outgoing eges put a 1 in t&e at& matri

"oing:

=in!lueJstio.&L=in!lueJ!onio.&L

int a?10@?10@A

voi main9B

int i-"nA

!lrs!r9A

printf>enter t&e no of verti!es n>9As!anf>F>Gn9A

printf>enter t&e a-a!en!y matri n>9A

foriH1AiJHnAi9
http://datastructures.itgo.com/graphs/adjmat.htmhttp://datastructures.itgo.com/graphs/adjmat.htm


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for-H1A-JH1A-9

s!anf>F>Ga?i@?-@9A

for"H1A"JHnA"9foriH1AiJHnAi9

for-H1A-JHnA-9

a?i@?-@Ha?i@?-@WWa?i@?"@GGa?"@?i@Aprintf>n t t&e transitive !losure is n>9A

foriH1AiJHnAi9

B for-H1A-JHnA-9

printf>t F>a?i@?-@9A

printf>n>9A

get!&9A

DTDT3

enter t&e no of verti!es4

enter t&e a-a!en!y matri0 1 0 0

0 0 1 0

1 0 0 10 0 0 0

t&e transitive !losure is

0 1 0 0 0 0 1 0

1 0 0 1

0 0 0 0

+. Aim3


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To rite a program to implement ,i-"stras algorit&m

Des!ription3

Oet P XE9 be a grap& X:Xerti!es G E:Eges .(e &ave to !&oose one verte as sour!e

t&e problem aim is to fin out minimum istan!e beteen sour!e noe an all remaining noes.

T&is problem is t&e !ase of orere paraigm in Preey met&o.

T&is problem !an be implemente by an algorit&m !alle ,i-"astra)s algorit&m.

Algorithm3

MMP be a grap&

MMCost matri ?13n13n@ for t&e grap& P


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MM#HBset of verti!es t&at pat& alreay generate

MMOet X be sour!e verte

MMist?-@A1JH-JHn enotes istan!e beteen X an -

voi main9

Bfor i3H1 to n o

Bs?i@HfalseA MM initiali;e s it& n

ist?i@H!ost?vi@A MMefine istan!e

s?v@HtrueA MMput v in s

[email protected] MM,istan!e beteen v an v is 0

for num3H2 to n:1 o

B

pat&s from v MM!&oose u from among t&ose verti!es not in # su!& t&at ist?u@HminA

s?u@HtrueA

forea!& a-as!ent to u it& s?@Hfalse9

ifist?@List?u@!ost?u@9

t&enist?@Hist?u@!ost?u@A MMupate t&e istan!e

Sour!e "oe3

=in!lueJstio.&L

=in!lueJ!onio.&L

=efine infinity $2767

int !ost?20@?20@nist?20@s?20@a?20@?20@Avoi setata9Avoi getata9A

voi pat&int9A

voi setata9

B

int i-"Aprintf>nEnter number of noes3 >9A

s!anf>F>Gn9A

printf>Enter *-a!en!y atri3 >9A

foriH1AiJHnAi9for-H1A-JHnA-9

s!anf>F>Ga?i@?-@9A

foriH1AiJHnAi9for-H1A-JHnA-9

B

ifiHH-9!ost?i@?i@H0A


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else if a?i@?-@UH09

B

printf>nEnter !ost from F to F3 >i-9As!anf>F>G!ost?i@?-@9A

else !ost?i@?-@HinfinityA

voi getata9

B

int iAforiH1AiJHnAi9

ifist?i@HH$27679

printf>not rea!&able>9A

else printf> F>ist?i@9A

voi pat&int v9

Bint i-minuA

foriH1AiJHnAi9

B

s?i@H0Aist?i@H!ost?v@?i@A

s?v@H1Aist?v@H0A

foriH2AiJHnAi9

BminH$2767A

for-H1A-JHnA-9

ifs?-@HH0 GG ist?-@Jmin9

uH-As?u@H1A

for-H1A-JHnA-9

ifs?-@HH0 GG a?u@?-@HH19ifist?-@List?u@!ost?u@?-@9

ist?-@Hist?u@!ost?u@?-@A


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voi main9

B

int vA!lrs!r9A

setata9A

printf>nEnter t&e sour!e verte3 >9As!anf>F>Gv9A

pat&v9A

printf>n#&ortest pat&s > 9Agetata9A

get!&9A

Output3

Enter number of noes3 6

Enter *-a!en!y atri3 0 1 1 1 0 0

0 0 1 1 0 00 0 0 0 1 0

1 0 0 0 1 00 1 1 0 0 0

0 0 0 0 1 0

Enter !ost from 1 to 23 +0

Enter !ost from 1 to $3 4+

Enter !ost from 1 to 43 10

Enter !ost from 2 to $3 10

Enter !ost from 2 to 43 1+

Enter !ost from $ to +3 $0


Enter !ost from 4 to +3 1+

Enter !ost from + to 23 20

Enter !ost from + to $3 $+

Enter !ost from 6 to +3 $

Enter t&e sour!e verte3 1

#&ortest pat&s 0 4+ 4+ 10 2+not rea!&able


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#esult3 Yen!e ,i-"stras algorit&m is implemente

6. Aim3 To Find the minimum cost spanning tree of a given undirectedgraph using Prim's algorithm.

Algorithm:

PrimG9

MM Input3 ,ire!te eig&te grap& G

MM Output3 inimum #panning Tree TT

U N B1 MM !onsier verte 1 as starting verte

$hile U Z V olet u v9 be t&e lo$est !ost ege su!& t&at

u U an v V : UTN T B u v9 MM a ege to spanning treeUN U B v

return T

enPrim.

Program3

=in!lueJstio.&L

=in!lueJ!onio.&L

intn !ost?10@?10@A

%oiprim9B

inti-"lnr?10@tempmin[!ostH0tree?10@?$@A

MR Vor first smallest ege RM

tempH!ost?0@?0@AforiH0AiJ nAi9

B

for-H0A-J nA-9

B iftempL!ost?i@?-@9

B tempH!ost?i@?-@A

"HiA

lH-A


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MR Qo e &ave fist smallest ege in grap& RM

tree?0@?0@H"Atree?0@?1@HlA

tree?0@?2@HtempA

min[!ostHtempA

MR Qo e &ave to fin min is of ea!&

verte from eit&er " or lby initialising nr?@ array

RM

foriH0AiJ nAi9 B

if!ost?i@?"@J !ost?i@?l@9

nr?i@H"A

else nr?i@HlA

MR To ini!ate visite verte initialise nr?@ for t&em to 100 RM

nr?"@H100A

nr?l@H100AMR Qo fin out remaining n:2 eges RM

tempH''A

foriH1AiJ n:1Ai9

B for-H0A-J nA-9

B

ifnr?-@UH100 GG !ost?-@?nr?-@@ J temp9 B

tempH!ost?-@?nr?-@@A

H-A

MR Qo i &ave got net verte RM

tree?i@?0@HAtree?i@?1@Hnr?@A

tree?i@?2@H!ost?@?nr?@@A

min[!ostHmin[!ost!ost?@?nr?@@Anr?@H100A

MR Qo fin if is nearest to any vertet&an its previous near value RM

for-H0A-J nA-9

B


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2$

ifnr?-@UH100 GG !ost?-@?nr?-@@ L !ost?-@?@9

nr?-@HA

tempH''A

MR Qo i &ave t&e anser -ust going to print it RMprintf>n T&e min spanning tree is3: >9A

foriH0AiJ n:1Ai9

B for-H0A-J $A-9

printf>F> tree?i@?-@9A

printf>n>9A

printf>n in !ost3: F> min[!ost9A

%oimain9

Binti-A

!lrs!r9A

printf>n Enter t&e no. of verti!es3: >9A

s!anf>F> Gn9A

printf >n Enter t&e !osts of eges in matri form3: >9AforiH0AiJ nAi9

for-H0A-J nA-9

s!anf>F>G!ost?i@?-@9A

printf>n T&e matri is3: >9A

foriH0AiJ nAi9 B

for-H0A-J nA-9

printf>Ft>!ost?i@?-@9A

printf>n>9A

prim9A

get!&9A

Output :

Enter t&e no. of verti!es3: $

Enter t&e !osts of eges in matri form3:

'' 2 $


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24

2 '' +

$ + ''

T&e matri is3:

'' 2 $

2 '' +$ + ''

T&e min spanning tree is3:0 1 2

2 0 $

in !ost3: +

7. Aim3 To implement Knapsa!" problem using ,ynami! programming.

Algorithm3

Step &. In t&is step t&e problem is ivie into t&e sub:problems means t&at t&e !&ara!teri;e t&e

stru!ture of an optimal solution. Ea!& sub:problem is solve an "eep tra!" ea!& items value.

Step '.In t&is step t&e re!ursive solution is fin. In t&is step optimal solution is step fist is

efine re!ursively.

Step (.In t&is step !ompute t&e !ost of optimal solution. In "napsa!" problem !ost is maimum

benefits.to !al!ulate t&e !ost is using t&e algorit&m &i!& is !alle 0M1 Knapsa!" *lgorit&m.

Step ). In t&is step t&e fin out t&e optimal solution. Vor fining t&e optimal solution e use t&e

ba!"tra!"ing.

Program3

#include#includeint n,p[20,![20"int ma$int a,int %&return a>%(a:%")int *napsac*$int i,int rc&

if$i++n&return$![i


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2+

for$i+-"i


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26

if s r S?k@ d9 an s S?k 1@ d9

"?k@ N 0

SumoSu!s k 1 r S?k@9

en SumoSu!.#include#include

void main(){int s[20],d,sum=0,n,x[20],top=0,i,tot=0;clsc();pint!("n$nte the num%e o! values ");scan!("&d",'n);pint!("n$nte the values in ascendin ode ");!o(i=0;i=0){x[top]=x[top];sum=sums[x[top]];i!(sum==d){pint!("n");tot=tot;!o(i=0;id//top>=n){sum=sum+s[x[top]];i!(top>=){sum=sum+s[x[top+]];*top=top+;*else{top=top;x[top]=x[top+];

**i!(tot==0){pint!("not possi%le ");*

etch();*


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27

AIM

To solve traveling salesman problem using ynami! programming.

AL*O#IT+M

1. ,e!lare a !lass ynami! it& ata members !ost matri number of !ities istan!e

array.

2. ,e!lare a private member fun!tion to fin out minimum of an array an publi!member fun!tions to getata9 isplay9 sap9 fa!t9 permutations an solutions.

$. getata9o 5ea member of !ities as n.

o Vor value of i from 1 to n by 1 repeat. Vor values of - from 1 to n by 1 repeat.

5ea !?i@?-@

o Vor values of I varying from 0 to fa!tn:19 by 1.

Oet ?i@H0

4. sapab9

o Oet tHa.

o aHb

o bHt

+. min9

o Oet mH?0@o Vor values of i to fa!tn:19 by 1.

C&e!" if mL?i@9 if so

o 5eturn m

6. sol9o Vor values of I from 0 to n:2 by 1 repeat.

Oet b?i@Hi2

o Call permutation fun!tion it& bon:2 as parameters.

7. fa!t9

o Oet fH1.

o Vor values of I from 1 to m by 1 repeat. fHfRi

o 5eturn f

/. isplay9o Vor values of I from 0 to fa!tn:19 by 1 repeat

rint ?i@

o rint min9


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2/

o Dsing a &ile loop -Jfa!tn:19 print t&e pat& also.

'. permlist?@ " m9

o C&e!" if "Hm if so K1H1

Vor values of I from 0 to m by 1 repeat.

at&?i@?i@Hlist?i@

?i@H !?"1@?list?i@@

"1Hlist?i@

?i@H?i@!?"1@?1@

let iHi1

o else

for values of i from " to m by 1 repeat.

Call saplist?"@ list?i@9

Call permlist "1 m9

Call saplist?"@ list?i@9

P#O*#AM

MMTravelling #alesman problem

=in!lue Jiostream.&L

=in!lue J!onio.&L

!lass ynami!Bprivate3

int !?+@?+@n?24@p?24@?6@list?+@rA

publi!3ynami!9A

voi getata9A

voi isplay9Aint fa!tint num9A

int minint list?@9A

voi permint list?@ int " int m9A

voi sol9AA

ynami!33ynami!9

BrH0A


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2'

voi ynami!33getata9B

!outJJ>Enter no. of !ities3>A

!inLLnA!outJJenlA

for int iH0AiJnAi9

for int -H0A-JnA-9!?i@?-@H0A

for iH0AiJnAi9

B

for -H0A-JnA-9B

if iUH-9

B

if !?i@?-@HH09B

!outJJ>Enter !ost from >JJiJJ> to >JJ-JJ> 3>A!inLL!?i@?-@A

!?-@?i@H!?i@?-@A

for iH0AiJn:1Ai9list?i@Hi1A

int ynami!33fa!tint num9

B

int fH1Aif numUH09

for int iH1AiJHnumAi9

fHfRiA

return fA

voi ynami!33permint list?@ int " int m9B

int itempA

if "HHm9B

for iH0AiJHmAi9

B

p?r@?i1@Hlist?i@A


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$0

rA

else

for iH"AiJHmAi9

B tempHlist?"@A list?"@Hlist?i@A list?i@HtempA

permlist"1m9A

tempHlist?"@A list?"@Hlist?i@A list?i@HtempA

voi ynami!33sol9B

permlist0n:29A

for int iH0AiJfa!tn:19Ai9B

p?i@?0@H0Ap?i@?n@H0A

for iH0AiJfa!tn:19Ai9B

?i@H0A

for int -H0A-JnA-9

B?i@H?i@!?p?i@?-@@?p?i@?-1@@A

int ynami!33minint list?@9B

int minimumHlist?0@A

for int iH0AiJfa!tn:19Ai9

Bif list?i@Jminimum9

minimumHlist?i@A

return minimumA

voi ynami!33isplay9


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$1

B

int i-A

!outJJenlJJ>T&e !ost atri3>JJenlAfor iH0AiJnAi9

B

for -H0A-JnA-9!outJJ!?i@?-@JJ>t>A

!outJJenlA

!outJJenlJJ>T&e ossible pat&s an t&eir !orresponing !ost3>JJenlA

for iH0AiJfa!tn:19Ai9

B

for -H0A-Jn1A-9!outJJp?i@?-@JJ>t>A

!outJJ>::L >JJ?i@JJenlA

!outJJenlJJ>T&e s&ortest pat& 3>JJenlAfor iH0AiJfa!tn:19Ai9

Bif ?i@HHmin99

brea"A

for -H0A-JHnA-9

B

!outJJp?i@?-@JJ> >A

!outJJenlJJ>nT&e !ost of t&is pat& is >JJ?i@JJenlA

voi main9

B

!lrs!r9Aynami! tsA

ts.getata9A

ts.sol9A

ts.isplay9Aget!&9A


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$2

OUTPUT

Enter no. of !ities34


Enter !ost from 0 to 2 3$0Enter !ost from 0 to $ 37+

Enter !ost from 1 to 2 3/0

Enter !ost from 1 to $ 320Enter !ost from 2 to $ 3'0

T&e !ost atri3

0 70 $0 7+

70 0 /0 20$0 /0 0 '0

7+ 20 '0 0

T&e ossible pat&s an t&eir !orresponing !ost3

0 1 2 $ 0 ::L $1+0 1 $ 2 0 ::L 210

0 2 1 $ 0 ::L 20+

0 2 $ 1 0 ::L 210

0 $ 2 1 0 ::L $1+0 $ 1 2 0 ::L 20+

T&e s&ortest pat& 30 2 1 $ 0

T&e !ost of t&is pat& is 20+


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$$

10. Aim3 To implement #trassen)s atri ultipli!ation.

Algorithm31. Consider multiplying two 2x2 matrices, as follows:

@ A 6 F + @AB @FAC

D E B C DEB DFEC

The obvious way to compute the right side is just to do the multiplies and ! additions. "ut

imagine multiplies are a lot more expensive than additions, so we want to reduce the number of

multiplications if at all possible. #trassen uses a tric$ to compute the right hand side with one

less multiply and a lot more additions %and some subtractions&.

2. 'ere are the ( multiplies:

4- + $@ E& 6 $ C& + @ @C E EC

42 + $@ A& 6 C + @C AC

4; + $D E& 6 + D E

4 + @ 6 $F C& + @F @C

4 + E 6 $B & + EB E

4 + $D @& 6 $ F& + D DF @ @F

4= + $A E& 6 $B C& + AB AC EB EC

). #o to compute *+"-, start with 1( %which gets us the *+ and "- terms&, then add/subtract

some of the other s until *+"- is all we are left with. iraculously, the 0s are chosen so that

1(2 wor$s. #ame with the other ) results re3uired.

!. 4ow just reali5e this wor$s not just for 2x2 matrices, but for any %even& si5ed matrices where the

*..' are submatrices.

Program3


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$4

=in!lueJstio.&L

int main9B

int a?2@?2@b?2@?2@!?2@?2@i-Aint m1m2m$m4m+m6m7A

printf>Enter t&e 4 elements of first matri3 >9AforiH0AiJ2Ai9

for-H0A-J2A-9

s!anf>F>Ga?i@?-@9A

printf>Enter t&e 4 elements of se!on matri3 >9A

foriH0AiJ2Ai9

for-H0A-J2A-9s!anf>F>Gb?i@?-@9A

printf>nT&e first matri isn>9A

foriH0AiJ2Ai9Bprintf>n>9A

for-H0A-J2A-9printf>Ft>a?i@?-@9A

printf>nT&e se!on matri isn>9A

foriH0AiJ2Ai9B

printf>n>9A

for-H0A-J2A-9printf>Ft>b?i@?-@9A

m1H a?0@?0@ a?1@?1@9Rb?0@?0@b?1@?1@9A

m2H a?1@?0@a?1@?1@9Rb?0@?0@A

m$H a?0@?0@Rb?0@?1@:b?1@?1@9Am4H a?1@?1@Rb?1@?0@:b?0@?0@9A

m+H a?0@?0@a?0@?1@9Rb?1@?1@A

m6H a?1@?0@:a?0@?0@9Rb?0@?0@b?0@?1@9A

m7H a?0@?1@:a?1@?1@9Rb?1@?0@b?1@?1@9A

!?0@?0@Hm1m4:m+m7A

!?0@?1@Hm$m+A!?1@?0@Hm2m4A

!?1@?1@Hm1:m2m$m6A

printf>n*fter multipli!ation using n>9A

foriH0AiJ2Ai9B

printf>n>9A

for-H0A-J2A-9


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$+

printf>Ft>!?i@?-@9A

return 0A

#ample output3

Enter t&e 4 elements of first matri3 1

2$

4

Enter t&e 4 elements of se!on matri3 +

67

/

T&e first matri is

1 2$ 4

T&e se!on matri is

+ 6

7 /

*fter multipli!ation using

1' 22

4$ +0


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$6

algorithms lab manual

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