he Completion of Training

N J has successfully completed Cpp test organized at with course material provided by the Spoken Tutorial

ducted remotely from IIT Bombay, is a pre-requisite for

g this training.

gineering College invigilated this examination. This

oken Tutorial Project, IIT Bombay. Score: 80.00%

ALGIORIT

ASSIG

Karatsuba A

JUny Ovide an

THMICS

GNMENT

Agnilihm

nd Senqyuel

INDE S.NO

TITLE INTRODUCTION

2

J

HISTORY

ALGOR ITHM

PROBLEM

IMPLEMENTAT 4ON

ExPLANA TION

AND

COMPLEx ITY

TIME

EX WRITTEN BY

PAGIE NO

KAYATRI -S -2115030

JOSHUA

SAMUEL -

ANANTHA kAA

P

2115028

PUSHPA

5

LESVWARANT- 21502 5

JEFFREY

JOS HwA

AMALANF215027

KAUSALYA -

IN ISHA

13

P 2115029

SALLOVE

R

-2115026

I1

Q

2

B 8

T

2

Hiotoy The standavd

procedune fov m

umbers vequhes a umev of

to nor 0n) in Hq -0 noto Lonfectwned thot the trodkional a

Yeouine Nn*) elemenkory o In 19bo, kolmoqoYov oroairzed n ykernetics a the

Proolems

he stabed the

(*) tonjech

the Complexity of Computak kovatzuba, then a23-yeor an

that

algorith

mulnplu

O 9 ) eleentary W

Conjectune. kelmogoTY duiscoveny: he uns h ch twhi r o Semi the

Communicoted

of

Gave

k on Bo Some lechures

Conterenes

all

ever

the

ntærotbo (ocecdi ngt efe tStockhelm 19b2 and pub

in the Siences

ro tealirgs o he

Kolmoojs rov and

tha

article catained

mukipic obia, karabuls

h

y

mulliplicokion of uo n -dig

of elemeptawy opevatlons pProptianad

obipr Andey komogorov

ahorithm for the task would opevotis a

Senuna7 on mathemaicod Mo8cou State Chiversiy, where and other pebems in

hre

week aki'on. Witin a r old student, fourd ues tuwo n-cligt nunes

steps

thus

olispreving

ta

alout the very extad

d

the athe.

ik

hers

next moebing meebing next

tevminated.

kavobaubo

Kolmeoprov

toyult ar

old (see For exaple,

congie of Mathemaiians blit hed the method n 1962

ora

a

USSR Acodemy of

hond buro

beap

toitten bly

reukts

odaonthm

and

a sepavole

ari Oneun

Yeult by Cfman "

k auhors the Poper

the

aunre

the

t

TepYinG

om

t

Example To

Compute te preduct

610 and

m-3. Then

the resultig base 6

de l00o)

we

1234S 12 loco+ 618 9 6

Only thre

untegev,are

coot

mulbplic alion

sed to comp

Z2 12xL =72 6 31 789- 2 12205

Z12-4245x+18 )-

3 6 1 x795-2-2722

A8 28 5-12-27290 5

the esulk bu Potal Jesuls, shifte We

get

by d in (ooo ko for

CaTries tto koase

accout

Ehe

uSuls=72 1oco +272205 = %39) e2

listed.

kara buba when

A karals.ko, and Ya ony become Te cei ve

he

th publisher.

t

o

12315 an

b72a,ehoose

ecomee the înput operands using ), as

+34S

t78

ns, uwhi th opevate on small

pute three pavtiad reslk .

5

-Z-Z0

205 =Il 538

ust adding bhese tree

accord1ngly and then taking

deconsesi 9bhece

npu cperands ). 1538

205

1000

thxee

înputs

Alaovhm

The algovitRm

princi

bosic

aloos

Pat

ovula

and

divide

s

log

Pvocuuct

ulfelicahon hol

about

each

plus

his

sone

sep

bosiC

ado S,

n

Comuplec

Similav

imoqmavy

usreve POuUev

Lot

diqt posihve

and

stigs

y

b

Sm

wtaqpv

wvite

wneve

and

av

Kovatsuls

ple

CCnuev, uSG

d

o

ne

Compute

numbevs

g

ns

and

SmalQ humbevs, aS

digits

J

as

digit ipls

and ohon9 oct,a g.gnevol 2ahon nulhplicahon

unitiis

be

me

v

alqovithmn,

veplaced

by

epesented

Ose

T.fov Can

n,one numbeS

less

as

Can

"

Tho

pvodct

n

is

m

( m

2

A2

ot 9,)

2

here

v 2

ooY

Gomulae and

oboeved

muul bpl

at

to

Cho

Can

al

cahons

2o

2,

kvoun

weve

and

vq

e

COst

vd

t 4,-

X,ot

7,

9,47,,4o

,4 o

9

(xA) t o t 3,)9,-7 (x

)

(tS,)-X,9, -

( x * ) (st )-25 2

) T4

quive

ouv mulhplicahos

ovles abloge kavat eu ba Ccomputed in n

ee

Octva

oddihions

o

a

fo COm

o-oJo

9,-69o

7,

-oy

V%

chsevve

at

tha

D Mutly Karatauos 0

ollouimg

mathodd amd 3 2 9 1 2

26732

Solution 2t o

ID 26132

32912

digit

1,

Her

number

algenhm pa

Kanala Ja

Aem

muml

O eus

dinnela

bath

oO2 Y

nw

-2xh

Oo73 L-yh

Oto2

673

291

Oo13

=Thh6 Sap 2

xh =

6132x

2

Sap

2912

3584

Cxht)x (4h l ) (oo2

6732) x (oo 13+211

68 34 x 2

85

2o30a43

g

s

amd

mumle

y 6

diat

numur

muluplcatuorn auaus

o73212

3 2 4- a\

2 2-

12)

D

O

(

T

M

D

E o

J

int

a

m/ (powCto, hay))

int

b

m-/(in t) (pow (to

knt

e

nl(pow (to, haig)

ut

ol

tint

ac

int

bd

nt

abcd

=

Retu

-/o

n

( ent) (pow (

(a, ) kanat suba ( b, d kaatsuba

karatSuba ( a

ac

pow (to,

2

*

bd

int

maint)

int , Y

Sans ("4-d .d", k x, Rg) PrEnts ( "Vod kaiatsa ba

Tnput 56 78

2 34

Output: 1006 b 52

);

o, halp))

g)); to,

halß)) ,

);

d); +b, c+d)- a c - b a

*half )+

: a ( n. Y);

abcd

* pow(to, hall

Explanatien The that

Kanatsuwa

algorithm

tecwxevely

JoAeaks

numbew

tnte

is a

4

don

th

Sub - pv blem

Smaule

omula mxn

( o An

whee m

*ac)

a,b,c d

The

En an

( lo ^ ( n / 2 ) * ( C a + b ) * c

ae nj2

digit

ruu

hal

En

and

+

us

undtton dice

used

te

&nteges un ctton

The

Stngle - dtqit

,

kac

t

check

hnumbe

t

Hetunns

thets

po

Aesult

as

multplication .

sHetwus

Elge, t

the

E =

and

5678

1 )hal

y=i234

k/2 = 2 .

i) a

5618/(o " 2)

iV) b

5618"/. (I0 2) 18

v)

34 /(to " 2)

C 2 2

V

1234'/. (10"2)

3H

multiplication algortthm

4ast

he

multeptication

size nl2

mk

Ct d )

-ac

umbevy

-

bd)

+bd

spli ttin

umber

the

efthe

the

wstng

outauned by

COLunt

s

twe

m

digts

n

oduct

wwstng Standand

s

the

pe

kanatsuba

algoithm.

1) ac

8

k a r a ( a , c)

bd

= 6T2

QL52

kaaa ( b d)

a) abe d

= k a r a l a + b , C+ d

to)x y

) -

ac-

bd

= ac* pow ( 10, 2 * halt) +abed TOO G6552

Tfme domplextty: TCn) =

3T(n 2)

whene

n

ina

the

the

Mastoi

Tin)= a

numben

Theoiem,

2 , n ) =On)

o(n

t n )40n) T(n)

)

algontthm

odta

T(nlb) + ftn)

a= 3, b

O(n Oln

+OCn)

Tin)

then

oln 585

=0(n72)

hastea than that

has

a

the

time

=I63 4

d

pow Cto , h a ) +bd

ait

the únput numbey

) =9(n *7b)

5) StandadA

multiplicatton

wmplex(ty Otn)

Certificate for th Cpp T

This is to certify that JEFFRY JOSHUVA AMALAN National Engineering College by Cammillus S

Project, IIT Bombay. Passing an online exam, cond

completing

Dheenathayalan Srinivasan from National Eng

training is offered by the Spo Credits: 2

March 22nd 2023

31100351JI