3.Growth of Functions

2

3.1 Asymptotic notation

g(n) is an asymptotic tight bound for f(n).

``=’’ abuse

}all for

)()()(0 s.t. ,,|)({))((

0

21021

nn

ngcnfngcnccnfng

))(()( ngnf

3

The definition of required every member of be asymptotically nonnegative.

n 0

n

c 2g(n)

c 1g(n)

f(n)f n g n( ) ( ( ))

4

Example:

)(6

.7 2

3214

23

222

nn

n>ifn

nnn

f n an bn c a, b, c a>

f(n)= (n ).

( ) ,

2

2

0 constants, .

In general,

).(nP(n)

aananp

d

didi

ii

Then

.0ith constant w are where)( 0

5

asymptotic upper bound} )()(0s.t.,|)({))(( 00 nnncgnfncnfngO

n 0

n

cg(n)

f(n)f n O g n( ) ( ( ))

6

asymptotic lower bound } )()(0 s.t. ,|)({))(( 00 nnnfncgncnfng

n 0

n

cg(n)

f(n)f n g n( ) ( ( ))

7

Theorem 3.1. For any two functions f(n) and g(n),

if and only if and

f n g n( ) ( ( ))

f n O g n( ) ( ( ))

f n g n( ) ( ( ))

8

o g n f n c n n n f n cg n( ( )) { ( )| , , ( ) ( )} 0 0 0

f n o g nf ng nn( ) ( ( )) lim( )( )

0

( ( )) { ( )| , , ( ) ( )}g n f n c n n n cg n f n 0 0 0

f n g nf ng nn( ) ( ( )) lim( )( )

9

Transitivity

Reflexivity

Symmetry

f n g n g n h n f n h n( ) ( ( )) ( ) ( ( )) ( ) ( ( )) f n O g n g n O h n f n O h n( ) ( ( )) ( ) ( ( )) ( ) ( ( )) f n g n g n h n f n h n( ) ( ( )) ( ) ( ( )) ( ) ( ( )) f n o g n g n o h n f n o h n( ) ( ( )) ( ) ( ( )) ( ) ( ( )) f n g n g n h n f n h n( ) ( ( )) ( ) ( ( )) ( ) ( ( ))

f n f n( ) ( ( ))f n O f n( ) ( ( ))f n f n( ) ( ( ))

f n g n g n f n( ) ( ( )) ( ) ( ( ))

10

Transpose symmetry f n O g n g n f n( ) ( ( )) ( ) ( ( )) f n o g n g n f n( ) ( ( )) ( ) ( ( ))

f n O g n a b( ) ( ( )) f n g n a b( ) ( ( )) f n g n a b( ) ( ( )) f n o g n a b( ) ( ( )) f n g n a b( ) ( ( ))

11

Trichotomy a < b, a = b, or a > b. e.g., does not always hold

because the latter oscillates between n n n, sin1

20nn

12

2.2 Standard notations and common functions

Monotonicity: A function f is monotonically increasing if m n im

plies f(m) f(n). A function f is monotonically decreasing if m n i

mplies f(m) f(n). A function f is strictly increasing if m < n implies f

(m) < f(n). A function f is strictly decreasing if m > n implies f

(m) > f(n).

13

Floor and ceiling x x x x x 1 1

n n n/ /2 2

n a b n ab/ / /

n a b n ab/ / /

bbaba /))1((/

bbaba /))1((/

For any integer n

For any real x

For any real n>=0, and integers a, b >0

14

Proof of Ceiling

1/01/0

/1/0/)1(/0

100 because*

done ,/// so,

1///

and ,1///

///

// then ,0 otherwise,

/// ,0 If

0 where,Let

/// :Prove

*

barsbar

abaraabar

abrabr

abnban

mabrmabrmabnright

mbsmbsmbsbm

ba

rbmb

a

rbmb

a

rabm

banleftr

mabnbanr

abrrabmn

abnban

15

Proof of Floor

1 /0

1/010 because*

done ,/// so,

,//)( and

,//)(//

////)(

// then ,0 otherwise,

,/// ,0 If

0 where,Let

/// :Prove

*

bars

barabr

abnban

mabrmabrabmright

mbsmbsbmbarbm

barbmbarabm

banleftr

mabnbanr

abrrabmn

abnban

16

Modular arithmetic For any integer a and any positive integer n, the v

alue a mod n is the remainder (or residue) of the quotient a/n :

a mod n =a - a/n n.

If(a mod n) = (b mod n). We write a b (mod n) and say that a is equivalent to b, modulo n.

We write a ≢ b (mod n) if a is not equivalent to b modulo n.

17

Exponential Basics

nmnm

mnmnnm

aaa

aaa

aa

aa

a

m, and nal aFor all re

)()(

/1

1

,0

1

1

0

18

Polynomials v.s. Exponentials Polynomials:

A function is polynomial bounded if f(n)=O(nk) Exponentials:

Any positive exponential function grows faster than any polynomial.

P n a ni i

i

d( )

0

n o a ab n ( ) ( )1

xnn

x

i

ix

en

x

xxxex

i

xe

)1(lim

1|| if 11

!2

0

19

Prove: nb=o(an)

nxn

n

a

n

xnxppn

nxC

xCxCxa

xaa

pbp

pp

p

n

b

ppppnpn

kn

k

nkn

kn

k

nk

nn

as 0)(

)()!1())!1((

!

)1(

),1( ,1 since

s.t.

11

1111)1(

00

20

Logarithm Notations

)lg(lglglg

)(lglg

logln

loglg 2

nn

nn

nn

nn

kk

e

21

Logarithm Basics

ac

ab

bb

c

cb

bn

b

ccc

a

bb

b

ca

ba

aa

b

aa

ana

baab

ba

, and n,, c, breal aallFor

loglog

log

log

1log

log)/1(log

log

loglog

loglog

loglog)(log

000

22

Logarithms

A function f(n) is polylogarithmically bounded if

for any constant a > 0. Any positive polynomial function grows faster

than any polylogarithmic function.

 

ln( ) ... | |12 3 4 5

12 3 4 5

x xx x x x

if x

1for )1ln(1

xxxx

x

k somefor )(lg)( nOnf k

)(log ab non

23

Prove: lgbn=o(na)

0lg

lim)2(

lglim

2lg

,0lim

lg

a

b

nna

b

n

a

n

b

n

n

nn

for and n for n, ang substituti

a

nBecause

24

Factorials Stirling’s approximation

  

))1

(1()(2!ne

nnn n

n o nn! ( )n n! ( ) 2

log( !) ( log )n n n

nn

ee

nnn

2!

where

nn n 12

1

112

1

25

Function iteration

.0))((

,0)( )1(

)(

ifｉnff

ifｉnnf i

i

For example, if , thennnf 2)( nnf ii 2)()(

26

The iterative logarithm function

nninin

in

n

)(i

)(i

iii

undefined. is lgor 0lg and 0 if undefined0lg and 0 if )lg(lg

0 if

)(lg

1

1

)1()1()(

lg ( ) min{ |lg }* ( )n i i 0 1

lg* 2 1lg* 4 2lg*16 3lg* 65536 4lg* 2 565536

h2..222* 2lg

h

27

Since the number of atoms in the observable universe is estimated to be about , which is much less than , we rarely encounter a value of n such that .

8010

655362

5lg* n

28

Fibonacci numbers

...61803.02

51

...61803.12

51

5

1

0

21

1

0

ii

i

iii

F

FFF

F

F

5

1 ,

5

1

12

51

2

511

00

2

51

2

51

2

51

011

1

0

22

BA

BAF

BAF

BAF

x

xxxxsolve

polynomialsticcharacteriUse

nn

i