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COSC 3101N J. Elder Thinking about Algorithms Abstractly Lecture 2. Relevant Mathematics: Classifying Functions Input Size Time f(i) = n (n)

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COSC 3101N

J. Elder

Lecture 2. Relevant Mathematics: Classifying Functions

Input Size

Tim

ef(i) = n(n)

COSC 3101N

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Assumed Knowledge

• Section 1.1 Existential and Universal Quantifiers

• Section 1.2 Logarithms and Exponentials

Loves( , )

Loves( , )

g b b g

g b b g

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Classifying Functions

Giving an idea of how fast a function grows without going into too much detail.

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Which are more alike?

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Which are more alike?

Mammals

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Which are more alike?

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Which are more alike?

Dogs

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Classifying Animals

Vertebrates

Birds

Mam

mals

Reptiles

Fish

Dogs

Giraffe

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Classifying Functions

T(n) 10 100 1,000 10,000

log n   3 6 9 13

n1/2 3 10 31 100

10 100 1,000 10,000

n log n 30 600 9,000 130,000

n2 100 10,000 106 108

n3 1,000 106 109 1012

2n 1,024 1030 10300 103000

Note: The universe contains approximately 1050 particles.

n

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Which are more alike?

n1000 n2 2n

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Which are more alike?

Polynomials

n1000 n2 2n

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Which are more alike?

1000n2 3n2 2n3

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Which are more alike?

1000n2 3n2 2n3

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Classifying Functions?

Functions

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Classifying Functions

Functions

Constant

Logarithm

ic

Poly L

ogarithmic

Polynom

ial

Exponential

Exp

Double E

xp

(log n)5 n5 25n5 log n5 2n5 25n

2

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Classifying Functions?

Polynomial

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Classifying Functions

Polynomial

Linear

Cubic

?

5n25n 5n3 5n4

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Logarithmic

• log10n = # digits to write n

• log2n = # bits to write n = 3.32 log10n

• log(n1000) = 1000 log(n)

Differ only by a multiplicative constant.

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Poly Logarithmic

(log n)5 = log5 n

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Logarithmic << Polynomial

For sufficiently large n

log1000 n << n0.001

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For sufficiently large n

10000 n << 0.0001 n2

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Polynomial << Exponential

For sufficiently large n

n1000 << 20.001 n

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Ordering Functions

Functions

Constant

Logarithm

ic

Poly L

ogarithmic

Polynom

ial

Exponential

Exp

Double E

xp

(log n)5 n5 25n5 log n5 2n5 25n

2

<< << << << << <<

<< << << << << <<

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Which Functions are Constant?

• 5• 1,000,000,000,000• 0.0000000000001• -5• 0• 8 + sin(n)

YesYesYes

No

YesYes

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Which Functions are “Constant”?

• 5• 1,000,000,000,000• 0.0000000000001• -5• 0• 8 + sin(n)

YesYesYesNo

NoYes

Lies in between

7

9

The running time of the algorithm is a “Constant” It does not depend significantly

on the size of the input.

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• n2

• … ?

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• n2

• 0.001 n2

• 1000 n2

Some constant times n2.

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• n2

• 0.001 n2

• 1000 n2

• 5n2 + 3n + 2log n

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• n2

• 0.001 n2

• 1000 n2

• 5n2 + 3n + 2log n

Lies in between

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• n2

• 0.001 n2

• 1000 n2

• 5n2 + 3n + 2log n

Ignore low-order termsIgnore multiplicative constants.

Ignore "small" values of n. Write θ(n2).

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Which Functions are Polynomial?

• n5

• … ?

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Which Functions are Polynomial?

• nc

• n0.0001

• n10000

n to some constant power.

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Which Functions are Polynomial?

• nc

• n0.0001

• n10000

• 5n2 + 8n + 2log n • 5n2 log n • 5n2.5

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Which Functions are Polynomial?

• nc

• n0.0001

• n10000

• 5n2 + 8n + 2log n • 5n2 log n • 5n2.5

Lie in between

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Which Functions are Polynomials?• nc

• n0.0001

• n10000

• 5n2 + 8n + 2log n • 5n2 log n • 5n2.5

Ignore low-order termsIgnore power constant.

Ignore "small" values of n. Write nθ(1)

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Which Functions are Exponential?

• 2n

• … ?

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Which Functions are Exponential?

• 2n

• 20.0001 n

• 210000 n

n times some constant powerraised to the power of 2.

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Which Functions are Exponential?

• 2n

• 20.0001 n

• 210000 n

• 8n

• 2n / n100 •2n · n100

too small?too big?

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Which Functions are Exponential?

• 2n

• 20.0001 n

• 210000 n

• 8n

• 2n / n100 •2n · n100

= 23n

> 20.5n

< 22n

Lie in between

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Which Functions are Exponential?

• 2n

• 20.0001 n

• 210000 n

• 8n

• 2n / n100 •2n · n100

= 23n

> 20.5n

< 22n

20.5n > n100

2n = 20.5n · 20.5n > n100 · 20.5n

2n / n100 > 20.5n

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Which Functions are Exponential?

Ignore low-order termsIgnore base.

Ignore "small" values of n. Ignore polynomial factors.

Write 2θ(n)

• 2n

• 20.0001 n

• 210000 n

• 8n

• 2n / n100 •2n · n100

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Classifying Functions

Functions

Constant

Logarithm

ic

Poly L

ogarithmic

Polynom

ial

Exponential

Exp

Double E

xp

(log n)θ(1) nθ(1) 2θ(n)θ(log n)θ(1) 2nθ(1) 2θ(n)

2

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Notations

Theta f(n) = θ(g(n)) f(n) ≈ c g(n)

BigOh f(n) = O(g(n)) f(n) ≤ c g(n)

Omega f(n) = Ω(g(n)) f(n) ≥ c g(n)

Little Oh f(n) = o(g(n)) f(n) << c g(n)

Little Omega f(n) = ω(g(n)) f(n) >> c g(n)

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

f(n) = θ(g(n))

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Definition of Theta

f(n) is sandwiched between c1g(n) and c2g(n)

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

f(n) = θ(g(n))

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Definition of Theta

f(n) is sandwiched between c1g(n) and c2g(n)

for some sufficiently small c1 (= 0.0001)

for some sufficiently large c2 (= 1000)

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

f(n) = θ(g(n))

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Definition of Theta

For all sufficiently large n

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

f(n) = θ(g(n))

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Definition of Theta

For all sufficiently large n

For some definition of “sufficiently large”

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

f(n) = θ(g(n))

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

c1·n2 3n2 + 7n + 8 c2·n2? ?

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n23 4

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·12 3·12 + 7·1 + 8 4·121False

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·72 3·72 + 7·7 + 8 4·727False

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·82 3·82 + 7·8 + 8 4·828True

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·92 3·92 + 7·9 + 8 4·929True

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·102 3·102 + 7·10 + 8 4·10210True

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n2?

COSC 3101N

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n2n 88

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n2

Truen 8

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n2

7n + 8 1·n2

7 + 8/n 1·n

n 8

True

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n2)

3·n2 3n2 + 7n + 8 4·n23 4 8 n 8

True

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

n2 = θ(n3)

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

n2 = θ(n3)

c1 · n3 n2 c2 · n3? ?

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

n2 = θ(n3)

0 · n3 n2 c2 · n30True, but ?

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Definition of Theta

c c n n n1 0 2 0 0 0, , , , . . .

n2 = θ(n3)

0 Constants c1 and c2 must be positive!

False

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

c1·n 3n2 + 7n + 8 c2·n? ?

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·n 3n2 + 7n + 8 100·n3 100

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·n 3n2 + 7n + 8 100·n?

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·100 3·1002 + 7·100 + 8 100·100100

False

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·n 3n2 + 7n + 8 10,000·n3 10,000

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Definition of Theta

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·10,000 3·10,000 2 + 7·10,000 + 8 100·10,00010,000

False

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Definition of Theta

3n2 + 7n + 8 = θ(n)

What is the reverse statement?

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

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Understand Quantifiers!!!

Sam Mary

Bob Beth

JohnMarilyn Monroe

Fred Ann

Sam Mary

Bob Beth

JohnMarilyn Monroe

Fred Ann

b, Loves(b, MM)

b, Loves(b, MM)

b, Loves(b, MM)]

b, Loves(b, MM)]

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

The reverse statement

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3n2 + 7n + 8 > 100·n3 100 ?

COSC 3101N

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·1002 + 7·100 + 8 > 100·1003 100

True100

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3n2 + 7n + 8 > 10,000·n3 10,000 ?

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·10,0002 + 7·10,000 + 8 > 10,000·10,0003 10,000

True10,000

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3n2 + 7n + 8 > c2·nc1 c2 ?

COSC 3101N

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·c22 + 7 · c2 + 8 > c2·c2c1 c2

Truec2

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3n2 + 7n + 8 > c2·nc1 c2 ?n0

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Definition of Theta

c c n n n c g n f n or f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( ) ( )

3n2 + 7n + 8 = θ(n)

3·c22 + 7 · c2 + 8 > c2·c2c1 max(c2,n0 )

Truec2 n0

True

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Definition of Theta

3n2 + 7n + 8 = g(n)θ(1)

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

3n2 + 7n + 8 = nθ(1)

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Definition of Theta

3n2 + 7n + 8 = nθ(1)

nc1 3n2 + 7n + 8 nc2

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

? ?

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Definition of Theta

3n2 + 7n + 8 = nθ(1)

n2 3n2 + 7n + 8 n3

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

2 3

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Definition of Theta

3n2 + 7n + 8 = nθ(1)

n2 3n2 + 7n + 8 n3

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

2 3 ? n ?

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Definition of Theta

3n2 + 7n + 8 = nθ(1)

n2 3n2 + 7n + 8 n3

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

2 3 5 n 5

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Definition of Theta

3n2 + 7n + 8 = nθ(1)

n2 3n2 + 7n + 8 n3

True

True

c1, c2, n0, n n0 g(n)c1 f(n) g(n)c2

2 3 5 n 5

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Order of Quantifiers

n n n c c c g n f n c g n0 0 1 2 1 2, , ( ) ( ) ( ), ,

f(n) = θ(g(n))

?

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

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g b loves b g, , ( , ) b g loves b g, , ( , )

Understand Quantifiers!!!

One girl Could be a separate girl for each boy.

Sam Mary

Bob Beth

JohnMarilin Monro

Fred Ann

Sam Mary

Bob Beth

JohnMarilin Monro

Fred Ann

COSC 3101N

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Order of Quantifiers

No! It cannot be a different c1 and c2

for each n.

n n n c c c g n f n c g n0 0 1 2 1 2, , ( ) ( ) ( ), ,

f(n) = θ(g(n))

c c n n n c g n f n c g n1 2 0 0 1 2, , , , ( ) ( ) ( )

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Other Notations

Theta f(n) = θ(g(n)) f(n) ≈ c g(n)

BigOh f(n) = O(g(n)) f(n) ≤ c g(n)

Omega f(n) = Ω(g(n)) f(n) ≥ c g(n)

Little Oh f(n) = o(g(n)) f(n) << c g(n)

Little Omega f(n) = ω(g(n)) f(n) >> c g(n)

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BigOh Notation

• n2 = O(n3)

• n3 = O(n2)

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BigOh Notation

• O(n2) = O(n3)

• O(n3) = O(n2)Odd Notation

f(n) = O(g(n)) Standard f(n) ≤ O(g(n)) Stresses one function dominating another.

f(n)O(g(n)) Stress function is member of class.