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