Asymptotic Notations for Time Efficiency Analysis

8/10/2019 Asymptotic Notations for Time Efficiency Analysis

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- INDIVIDU

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nya kirim ke email bu astuti yg ini [email protected]

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-KELOMPOK

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Asymptotic Notations forTime Efficiency Analysis

Design and Analysis of Algorithms


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Contents

Asymptotic Notations:

O(big oh)

(big omega)

(big theta)

Basic Efficiency Classes


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In the following

discussiont(n) & g(n): any nonnegative functions

defined on the set of natural numbers

t(n)an algorithms running timeUsually indicated by its basic operation count

C(n)

g(n)some simple function to compare

the count with


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O(g(n)): Informally

O(g(n))is a set of all functions with a

smalleror same order of growth as g(n)

Examples:

n O(n2); 100n + 5 O(n2)

n (n-1) O(n2)

n3O(n2); 0.0001 n3O(n2); n4+n+1 O(n2)


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(g(n)): Informally

(g(n))is a set of all functions with a

largeror same order of growth as g(n)

Examples:

n3(n2)

n (n-1) (n2)

100n + 5 (n2)


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(g(n)): Informally

(g(n))is a set of all functions with a

same order of growth as g(n)

Examples:

an2+bn+c; a>0 (n2); n2+sin n (n2)

n (n-1) (n2); n2+log n (n2)

100n + 5 (n2); n3(n2)

=


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O-notation: Formally

DEF1: A function t(n) is said to be in

O(g(n)), denoted t(n) O(g(n)), if t(n) is

bounded aboveby some constant multiple

of g(n) for all large n

i.e. there exist some positive constant c

and some nonnegative integer n0, suchthat

t(n) cg(n) for all n n0


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t(n) O(g(n)): Illustration

n

n0

doesn't

matter

t(n)

cg(n)


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Proving Example: 100n + 5 O(n2)

Remember DEF1: find c and n0, such that


100n + 5 100n + n (for all n 5) = 101n 101n2c=101, n0=5

100n + 5 100n + 5n (for all n 1) = 105n 105n2c=105, n0=1


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-notation: Formally


(g(n)), denoted t(n) (g(n)), if t(n) is

bounded belowby some constant multiple

of g(n) for all large n

i.e. there exist some positive constant c

and some nonnegative integer n0, suchthat



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t(n) (g(n)): Illustration

n

n0

doesn't

matter

t(n)

cg(n)


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Proving Example: n3

(n2)

Remember DEF2: find c and n0, such that


n3 n2 (for all n 0)c=1, n0=0


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-notation: Formally


(g(n)), denoted t(n) (g(n)), if t(n) is

bounded both aboveand belowby some

constant multiple of g(n) for all large n

i.e there exist some positive constant c1

and c2and some nonnegative integer n0,such that

c2g(n) t(n) c1g(n) for all n n0


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t(n) (g(n)): Illustration

nn0

doesn't

matter

t(n)

c1g(n)

c2g(n)


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Proving Example: n(n-1) (n2)

Remember DEF3: find c1and c2and some

nonnegative integer n0, such that

c2g(n) t(n) c1g(n) for all n n0

The upper bound: n(n-1) = n2 n n2 (for all n 0)

The lower bound: n(n-1) = n2 n n2- n n (for all n 2) = n2

c1= , c

2= , n

0= 2


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Basic Efficiency Classes

The time efficiency of a large number of

algorithms fall into only few classes

1, log n, n, n log n, n2, n3, 2n, n!

Multiplicative constants are ignoredit is

possible that an algorithm in worse

efficiency class running faster than

algorithm in better class

Exp: Alg A: n3, alg B: 106n2; unless n > 106,

alg B runs faster than alg A


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Class: 1

Name: constant

Comment:

Short of best-case efficiency

Very few reasonable examples can be given

algs running time typically goes to infinitywhen its input size grows infinitely large


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Class: log n

Name: logarithmic

Comment:

Typically, a result of cutting a problems sizeby a constant factor on each iteration of the

algorithm

Logarithmic alg cannot take into account all its

input or even a fixed fraction of it: anyalgorithm that does so will have at least linear

running time


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Class: n

Name: linear

Comment:

Algorithms that scan a list of size n (e.g.

sequential search) belong to this class


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Class: n2

Name: quadratic

Comment:

Typically, characterizes efficiency of

algorithms with two embedded loops

Standard examples: elementary sorting

algorithms and certain operations on n-by-n

matrices


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Class: n3

Name: cubic

Comment:

Typically, characterizes efficiency of

algorithms with three embedded loops

Several nontrivial algorithms from linear

algebra fall into this class


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Class: 2n

Name: exponential

Comment:

Typical for algorithms that generate all

subsets of an n-element set

The term exponential is often used to includethis and faster orders of growth as well


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Class: n!

Name: factorial

Comment:

Typical for algorithms that generate all

permutations of an n-element set


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Useful Property

Theorem:

If t1(n) O(g1(n)) and t2(n) O(g2(n)),

then t1

(n) + t2

(n) O(max{g1

(n), g2

(n)})

The analogous assertions are true for the

and

notations as well


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Example

Alg to check whether an array has

identical elements:

Sort the array

Scan the sorted array to check its

consecutive elements for equality

(1) = n(n-1) comparisonO(n2)

(2) = n-1 comparisonO(n)The efficiency of (1)+(2) = O(max{n2,n})

= O(n2)


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Using Limits for Comparing

OoG

A convenient method for comparing order ofgrowth of two specific functions

Three principal cases:

The first two cases t(n) O(g(n)); the last twocasest(n) (g(n)); the second case alone

t(n) (g(n))

gOolargahasthat t(implies

gaOosamthehasthat t(impliesc

Osmaahasthat t(implies0

)(

)(limng

nt

n


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Limit-based: why

convenient?It can take advantage of the powerful

calculus techniques developed for

computing limits, such as

LHopitals rule

Stirlings formula

('

('lim

)(

)(lim

g

t

ng

nt

nn

ovlargfor2! ne

n

nn


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Example (1)

Compare OoG of n(n-1) and n2.

The limit = cn(n-1) (n2 )

Compare OoG of log2n and n

The limit = 0log2n has smaller order of n

1li2

1lim2

1)1(lim 1

2

2

2

21

n

nnn n

nn

n

nn

lilo2)(loglim)'(

)'(loglimloglim 22

1

1222

eennnn

nnnnnn


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Example (2)

Compare OoG of n! and 2n.

The limit = n! (2n )

n

nnn

n

nn

n

e

n

nn

n en

ennnn 22li

22lim

2

2lim2

!lim


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Exercises (1)

True or false:

n(n+1)/2 O(n3)n(n+1)/2 O(n2)

n(n+1)/2 (n3)n(n+1)/2 (n)

Indicate the class (g(n)):

(n2

+1)10

(10n2+7n+3)

2n log (n+2)2+(n+2)2log (n/2)


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Exercises (2)

1. Prove that every polynomial

p(n) = aknk+ ak-1n

k-1+ + a0with ak> 0belongs to (nk)

Prove that exponential functions anhavedifferent orders of growth for different

values of base a > 0

Asymptotic Notations for Time Efficiency Analysis

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