Asymptotic Notation and Analysis in Algorithm Complexity, Slides of Data Structures and Algorithms

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Data Structures and Algorithms

CSE- 101

• asymp - 1 Comp

Asymptotic Notation

Asymptotic Analysis

 Asymptotic analysis is a method of

describing limiting behavior.

 Suppose that we are interested in the properties

of a function f ( n ) as n becomes very large.

 f ( n ) = n

2

+ 3 n ,

 As n becomes very large, the term 3 n becomes

insignificant compared to n

2

Asymptotic Complexity

 Running time of an algorithm as a function of

input size n for large n.

 Expressed using only the highest-order term in

the expression for the exact running time.

 Instead of exact running time, say Q( n

2

 Describes behavior of function in the limit.

 Written using Asymptotic Notation.

Asymptotic Notation

 Q , O , W , o , w

 Defined for functions over the natural numbers.

 Ex: f ( n ) = Q( n

2

 Describes how f ( n ) grows in comparison to n

2

 Define a set of functions; in practice used to compare

two function sizes.

 The notations describe different rate-of-growth

relations between the defining function and the

defined set of functions.

Q-notation

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

 positive constants c 1 , c 2 , and n 0, such that  n  n 0 , we have 0  c 1 g ( n )  f ( n )  c 2 g ( n )

For function g ( n ), we define Q( g ( n )), big-Theta of n , as the set: Technically, f ( n )  Q( g ( n )). Older usage, f ( n ) = Q( g ( n )). f ( n ) and g ( n ) are nonnegative, for large n****.

Example

 10 n

2

• 3 n = Q( n 2

 What constants for n

0

, c

1

, and c

2

will work?

 Make c

1

a little smaller than the leading

coefficient, and c

2

a little bigger.

 To compare orders of growth, look at the

leading term.

 Exercise: Prove that n

2

/2- 3 n = Q( n

2

Q ( g ( n )) = { f ( n ) :  positive constants c

1 , c 2 , and n 0

such that  n  n 0 , 0  c 1 g ( n )  f ( n )  c 2

g ( n )}

O - notation

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

 positive constants c and n 0, such that  n  n 0 ,

we have 0  f ( n )  c g ( n ) }

For function g ( n ), we define O ( g ( n )), big-O of n , as the set: g ( n ) is an asymptotic upper bound for f ( n ). Intuitively : Set of all functions whose rate of growth is the same as or lower than that of g ( n ). f ( n ) = Q ( g ( n ))  f ( n ) = O ( g ( n )). Q ( g ( n ))  O ( g ( n )).

Examples

 Any linear function an + b is in O ( n

2

). How?

 Show that 3 n

3

= O ( n

4

) for appropriate c and n

0

O ( g ( n )) = { f ( n ) :  positive constants c and n

0

such that  n  n

0

, we have 0  f ( n )  c g ( n ) }

Example

 n = W(lg n ). Choose c and n

0

W( g ( n )) = { f ( n ) :  positive constants c and n

0

, such

that  n  n

0

, we have 0  c g ( n )  f ( n )}

Relations Between Q, O, W

o - notation

f ( n ) becomes insignificant relative to g ( n ) as n

approaches infinity:

lim [ f ( n ) / g ( n )] = 0 n 

g ( n ) is an upper bound for f ( n ) that is not

asymptotically tight.

Observe the difference in this definition from previous

ones.

o ( g ( n )) = { f ( n ):  c > 0 ,  n

0

> 0 such that

 n  n

0

, we have 0  f ( n ) < cg ( n )}.

For a given function g ( n ), the set little- o :

w ( g ( n )) = { f ( n ):  c > 0 ,  n

0

> 0 such that

 n  n

0

, we have 0  cg ( n ) < f ( n )}.

w - notation

f ( n ) becomes arbitrarily large relative to g ( n ) as n

approaches infinity:

lim [ f ( n ) / g ( n )] = .

n 

g ( n ) is a lower bound for f ( n ) that is not

asymptotically tight.

For a given function g ( n ), the set little-omega: