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Growth of Functions and Asymptotic Notation

Overview

Study a way to describe behavior of functions in the limit ...

asymptotic efficiency Describe growth of functions

Focus on whats important by abstracting lower-order terms andconstant factors

How we indicate running times of algorithms

A way to compare sizes of functions

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O-notation

Asymptotic upper bound O-notation

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

if there exists constants c and n0 such that

0 f(n) c g(n) forn n0

say: g(n) is an asymptotic upper bound for f(n). Example:

2n + 10 =O(n2), pick c= 1 and n0 = 5

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More on O-notation

Thing ofO(g(n)) as a set of functions

O(g(n)) ={f(n) : c, n0

such that 0 f(n) c g(n) forn n0

} Examples of functions in O(n2):

n2 +nn2 + 1000n1000n2 + 1000n

n/1000n2/ lg n

O-notation is an upper-bound notation. Makes no sense to say f(n) isat leastO(g(n))? (why?)

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-notation

Asymptotic lower bound -notation

f(n) =(g(n))

if there exists constants c and n0 such that

0 c g(n) f(n) forn n0

say: g(n) is an asymptotic lower bound for f(n). Example:

n= (lgn), pick c= 1 and n0 = 16

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More on -notation

Thing of(g(n)) as a set of functions

(g(n)) ={f(n) : c, n0

such that 0 c g(n) f(n) forn n0

} Examples of functions in (n2):

n2

n2 +nn2 n

1000n2 + 1000n1000n2 1000nn2.00001

n2 lg nn3

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-notation

Asymptotic tight bound -notation

f(n) =(g(n))if there exists constants c1, c2 and n0 such that

0 c1 g(n) f(n) c2g(n) forn n0

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

1

2n2 2n= (n2), pick c1 = 14 c2 =

1

2 and n0 = 8.

Ifp(n) =

d

i=1ain

i and ad >0, then p(n) =O(nd)

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More on -notation

Thing of(g(n)) as a set of functions

(g(n)) ={f(n) :

c1, c2, n0 such that 0

c1g(n)

f(n)

c2g(n) forn

Examples of functions in (n2):n2

n2 +nn2 n

1000n2 + 1000n1000n2 1000n

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Theorem

Theorem. O and iff.

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Using limits for comparing orders of growth

In order to determine the relationship between f(n) and g(n), it is oftenusefuly to examine

limn f(n)g(n) =L

The possible outcomes:

1. L= 0: f(n) =O(g(n))

2. L= : f(n) =(g(n))

3. L= 0 is finite: f(n) =(g(n))

4. There is no limit: this technique cannot be used to determine theasymptotic relationship between f(n) and g(n).

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Examples

1. f(n) =n2 and g(n) =n lgn

n2 =(n lg n)

2. f(n) =n100 and g(n) = 2n

n100 =O(2n)

3. f(n) = 10n(n+ 1) and g(n) =n2

10n(n+ 1) =(n2)

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