Data Structures and Algorithms
with Object-Oriented Design Patterns in C++ 
This section presents two less commonly used forms of asymptotic notation. They are:
, to describe a function which is
both O(g(n)) and 
, for the same g(n).
(Definition 
).
, to describe a function which is
O(g(n)) but not 
, for the same g(n).
(Definition ).
Definition (Theta) Consider a function f(n) which is non-negative for all integers. We say that ``f(n) is theta g(n),'' which we write
, if and only if f(n) is O(g(n)) and f(n) is
Recall that we showed in Section that
a polynomial in n,
say 
,
is 
.
We also showed in Section that
a such a polynomial is 
.
Therefore, according to Definition ,
we will write 
.
Definition (Little Oh) Consider a function f(n) which is non-negative for all integers
Little oh notation represents a kind of
loose asymptotic bound
in the sense that if we are given that f(n)=o(g(n)),
then we know that g(n) is an asymptotic upper bound
since f(n)=O(g(n)),
but g(n) is not an asymptotic lower bound
since f(n)=O(g(n)) and 
implies that 
.
For example, consider the function f(n)=n+1.
Clearly, 
.
Clearly too, 
,
since not matter what c we choose,
for large enough n, 
.
Thus, we may write 
.
Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.