2. Asymtotic Notation

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Algorithms


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REVIEW: RUNNING TIME

Number of primitive opertio!s thatare executed:

Except for time of executing afunction call most statementsroughly require the same amount of

time

We can be more exact if need be.


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AN E"AMPLE: INSERTIONSORTI!p#t: Arr$ A o% si&e !O#tp#t: Sorte' rr$ i! i!(resi!g or'er

I!sertio!Sort)A* !+ ,%or i - . to ! ,/e$ - A0i1

2 - i 3 456hile )2 7 8+ !' )A021 7 /e$+ ,A02941 - A021 2 - 2 3 4

A02941 - /e$


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INSERTION SORTStatement Eort

I!sertio!Sort)A* !+ ,%or 2 ; . to ! , c1n/e$ ; A021 c2n!1"i ; 2 3 45 c#n!1"

6hile )i 7 8+ !' )A0i1 7 /e$+ , c$ % &A0i941 ; A0i1 c' % &!n!1""

i ; i 3 4 c( % &!n!1""

)A0i941 ; /e$ c*n!1"

) % &+ t2 , t# , - , tn here t & is number of times youshifting the element in right side to insert ne number.


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ANALY


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ANALY?est (se @The best case occurs if the array is alreadysorted.

Therefore,t j = 1 for j = 2, 3, ...,n and the best-case running time

can be computed using above equation as follows:This running time can be expressed as An + B for

constants Aand B that depend on the statement costsci.

Therefore, T(n) it is a linear function ofn.


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ANALYWorst (se@The worst case occurs if the array is reversesorted.

Therefore,t j = 1 for j = 2, 3, ...,n and the worst-case running time

can be computed using above equation.This running time can be expressed as An 2 + Bn + Cforconstants A , B and C that depend on the statement

costsci.

Therefore, T(n) it is a quadratic function ofn.


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ASYMPTOTIC NOTATIONS


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ASYMPTOTIC PERORMANCE

Suppose e are considering t6o lgorithms* A !' B*for sol/ing a gi/en problem.

0unning times of each of the algorithms are %n" and %n"

respecti/ely3 here n is a measure of the problem si5e.

Whi(h is =etter Algorithm: 6f e 7no the problem si5e ! in ad/ance and then if

TA)!+ T?)!+ conclude that algorithm A is better thanlgorithm ?D

U!%ort#!tel$* 6e #s#ll$ 'o!t /!o6 the pro=lemsi&e =e%oreh!'D


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ASYMPTOTIC PERORMANCE

Asymptotic performance: 8o does anlgorithm beha/e as the problem input"si5e gets /ery large9

0unning time

emory;storage requirements


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ALGORITFMS WITF SAMECOMPLE"ITY

•0epresent


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ASYMPTOTIC NOTATION

Big-Oh O(.)Big-Oh O(.)

Big-OmegaΩ(.)

Big-ThetaΘ(.)

Little-Oh o(.)

Little-Omegaω(.)

> Fortunately, there are asymptoticnotations which allow us to characterizethe main factors aecting analgorithm’s running time without goinginto detail

> A good notation for large inputs.

> The notations describe dierent rate-of-

growth relations between the deningfunction and the dened set offunctions.


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ASYMPTOTIC UPPER ?OUNH3?IG OF


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ASYMPTOTIC UPPER ?OUNH3?IG OF

?et fn" and gn" represents complexities ofto algorithms.

fn" is said to be in the family of gn" i

8 ≤ %)!+ ≤ ( Dg)!+ %or ll ! J !8


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?IG O NOTATION

Emple 6f fn" + 2n2

gn" is @n# "

K#e: Sho6 tht %)!+ is ?ig Oh g)!+

A!s: to pro/e this e ha/e to sho that ) ≤ fn" ≤ c gn"

for all n A n)

6f e can Bnd positi/e constants ( !' !8 thene can say that %)!+ is =ig oh g)!+


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?IG O NOTATION

Emple 4 6f fn" + 2n2 , gn" is O(n3)

fn" ≤ c gn"

2n2

≤ cn3⇒

2 ≤ cn⇒

c = 1 and n0

= 2 So, for c=1 and for all n A 2, f(n) is big oh g(n)

There is no unique set of values forno and cin proving

the asymptotic bounds but for whatever value of no youtake, it should satisfy for all n > n

o

Or For c=2 and n

o =4 also, f(n) is big oh g(n)


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?IG O NOTATION

Emple . 6f fn" + 8n+128, gn" is O(n2)

%o pro/e fn" is big oh gn" e ha/e to pro/ethat

%)!+ ≤ ( g)!+ %or ll ! J !8 !' (o!st!t (Lets t/e (;4

Since (n+8)>0 for all values of! J 8 , so for!8;4 Above rule is saisfied! So, for c=1 and for all n J 1", f(n) is big oh g(n)


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?IG3O NOTATION

When determining the order of a functionfx"3 alays try to pic7 the smallest gx"possible.

6f fx" is Οgx"" and gx" is Οfx""3 then fx"and gx" are of the same order.

6n practice3 all big!@ results are obtained forfunctions that are positi/e for all /alues ofx.


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ASYMPTOTIC LOWER ?OUNH: 3NOTATION

ig @ is asymptotic upper bound

Ω!notation is asymptotic loer bound


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3NOTATION

family of functions e say `̀ f(n) is omega g(n),'' iff

C fn" : there exist positi/e constants c and no such

that

)≤ c gn"

≤ fn" for all nA n) D


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3NOTATION

Example:

=or any to functions fn" and gn" e ha/e

fn" + 5n2 -64n+256 and gn" + Ωn2"

so* %)!+ ; )g)!+ i% !' o!l$ i% (D g)!+ ≤ %)!+

Sol: in order to sho this e need to Bnd aninteger n) and a constant c!) such that

for all integers nA n) fn" A cn2


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3NOTATION

Example:

=or any to functions fn" and gn" eha/e

fn" + 5n2 -64n+256 and gn" + Ωn2" suppose e (hoose (;43 then

So here (n-8)2 Is always positive for all n ≥ 0 so n0= "

Hence for c=1 and n0 =0 ``f(n) is omega g(n)!!


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3 NOTATION HEINITION

Θ gn" " denotes a family of functions.

`̀ f(n) is Theta g(n),'' if e can Bndconstants c13 c23 and n) such that

) ≤ c1 gn" ≤ fn" ≤ c2 gn"for all n A n)


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=or any to functions fn" and gn" eha/e

fn" + Θgn""if and only if

fn" + @gn"" and fn" + Ωgn""


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ASYMPTOTIC TIGFT ?OUNH

%his is said to asymptotictight bound.

=or reasonably large

/alues of n3 the functionfn" is ithin the range ofconstant multiples ofgn"

fn" is bounded belo asell as abo/e.


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RELATIONS ?ETWEENNOTAIONS


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3 NOTATIONEmple6f fn" + *n2 '$n ,#

gn" is Θ n2 "

K#e: Sho6 tht %)!+ is Thet g)!+D

ns: 6f fn" is also ha/ing the same running timeas gn" then pro/e folloing rule: ) ≤ c1 gn" ≤

fn" ≤ c2 gn"

for all n A n)

No Bnd c13 c2 and n)

We want c c and n such


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7n2 – 54n +3We want c 1, c 2 and n0 such

that

c 1n2 ≤ 7n2 – 54n +3 ≤ c 2 n

2

for all n ≥ n0

Select c 1 " # and #" c 2

Say c 1 = 5 and c 2 = 9

o !et 5n2 ≤ 7n2 – 54n +3,

n "ust #e #$!!er than 27 %7n2 – 54n +3 ≤ 9n2 $s true for all n%

hus c 1 = 5, c 2 = 9 and n0 = 2&


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TAE ANOTFER E"AMPLE

6f fn" + *n# '$n2 , 1)n , #


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LITTLE O3NOTATION

ogn" +C fn":

for any positi/e constant c F )there exists a constant n) F ) such

that) G fn" H cgn" for all n ≥ n) D

?imit f;g → )


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LITTLE ω3NOTATION

ωgn" +C fn":

for any positi/e constant c F )there exists a constant n) F ) such

that) G cgn" H fn" for all n ≥ n) D

?imit g;f → )


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NAMES O MOST COMMON?IG OF UNCTIONS


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TFEOREMR4:R4: 6f dn" is @fn""3 then adn" is @fn"3 a

F )

R.:R.: 6f dn" is @fn"" and en" is @gn""3then dn",en" is @fn",gn""

R:R: 6f dn" is @fn"" and en" is @gn""3

then dn"en" is @fn"gn""

R:R: 6f dn" is @fn"" and fn" is @gn""3then dn" is @gn""


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TFEOREM

R:R: 6f fn" + a) , a1n , - , adnd3 d and a7 are constants3 then fn" @nd"

R:R: nx is @an" for any Bxed x F ) and a F 1

RQ:RQ: log nx is @log n" for any Bxed x F )

R:R: log xn is @ny" for any Bxed constants xF ) and y F )


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SOME FELPULMATFEMATICSConstant Series: For integersaandb,a≤ b,

Linear Series (Arithmetic Series): Forn ≥ 0,

Quadratic Series:Forn ≥ 0,

∑=

++=+++=

n

i

nnnni

1

2222

6

)12)(1(21

∑=

+−=b

ai

ab 11

2

)1(21

1

+=+++=∑

=

nnni

n

i


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SOME FELPULMATFEMATICS Cubic Series:Forn ≥ 0,

Geometric Series: For real x ≠ 1,

For | x| < 1,

∑=

+=+++=

n

i

nnni

1

223333

4

)1(21

∑=

+

−

−=++++=

n

k

n

nk

x

x x x x x

0

12

1

11

∑∞

= −=

0 1

1

k

k

x x


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SOME FELPULMATFEMATICS Cubic Series:Forn ≥ 0,

Geometric Series: For real x ≠ 1,

For | x| < 1,

∑=

+=+++=

n

i

nnni

1

223333

4

)1(21

∑=

+

−

−=++++=

n

k

n

nk

x

x x x x x

0

12

1

11

∑∞

= −=

0 1

1

k

k

x x

2. Asymtotic Notation

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