CSCI 248 Data Structures and Algorithms II
Growth of Functions, Asymptotic Notations (Chapter 3 of the textbook)
T(n) – running time function, n

{0,1,2,….}
O-notation

Definition : A function T(n) is said to be in O(g(n)) if there exist positive constants c and n o such that 0

T(n)

cg(n) for all n

n o
.

We write T(n)=O(g(n)).

O-notation describes an upper bound.
Examples:
1. Prove that 100n+5 = O(n
2
)
2. Prove that 3 n
 
( n log n )

- notation

Definition : A function T(n) is said to be in

(g(n)) if there exist positive constants c and n o such that 0

cg(n)

T(n) for all n

n o
.

We write T(n)=

(g(n)).
 
-notation describes an lower bound.
Example:
1.
Prove that n
3  
( n
2
)

- notation

Definition : A function T(n) is said to be in

(g(n)) if there exist positive constants c
1
, c
2,
and n o such that c
1 g(n)

T(n)

c
2 g(n) for all n

We write T(n)=

(g(n)).
 
-notation asymptotically bounds a function from above and below

n o
.
Example:
1. Prove that
1
2 n ( n

1 )
 
( n
2
)