The document explains the concepts of little oh (o) and little omega (ω) notations used to denote upper and lower bounds on the growth rates of algorithm running times, which may not be asymptotically tight. Little oh (o) defines an upper bound that is always less than a corresponding function, while little omega (ω) defines a lower bound that is always greater. It further provides definitions, differences between corresponding big O and big Omega notations, and examples of applying these concepts using limit formulas.

1. Lecture Note-11:Little oh(o) and Little omega(ω) 06 Feb 2016
By Rajesh K Shukla, HOD, Department of CSE, SIRTE Bhopal
Downloaded from www.RajeshkShukla.com
The little Oh (o) notation is a method of expressing the an upper bound on the growth rate of an algorithm’s
running time which may or may not be asymptotically tight therefore little oh(o) is also called a loose upper
bound we use little oh (o) notations to denote upper bound that is asymptotically not tight. little oh( o)can
formally be defined as follows
“Given functions f(n) and g(n), we say that f(n) is little oh of (g(n)) if there are positive constants c and n0
such that f(n) < cg(n) for all n, n ≥ n0 that is, f has a lower growth rate than g. So little oh (o) is to mean
“tight upper bound”. The main difference between Big Oh (O) and little oh (o) lies in their definitions. In Big
Oh f(n) = O(g(n)) and the bound is 0<=f(n)<=cg(n) so it is true only for some positive value of ‘c’ but in case
of little oh, it is true for all constant c>0 because is f(n)= o(g(n)) and the bound f(n)<cg(n). The little oh (o)
can also be defined as a functions of limits as given below
Example prove that 4n+6=o(n2
)
Solution
The Little oh (o) running time can be proved by applying the limit formula as given below
If
)(
)(
lim ng
nf
n 
=0, then functions f(n) is o(g(n))
Here we have
)(n)(and64n)( 2
 ngnf
So
0)
6
n
4
(
n
64n
)(
)(
limlimlim 2



 nng
nf
nnn
Therefore
4n+6=o(n2
)
The little omega (ω) notation is a method of expressing the an lower bound on the growth rate of an
algorithm’s running time which may or may not be asymptotically tight therefore little omega(ω) is also
called a loose lower bound we use little omega (ω) notations to denote lower bound that is asymptotically not
tight. little omega (ω)can formally be defined as follows
“Given functions f(n) and g(n), we say that f(n) is little omega of (g(n)) if there are positive constants c and
n0 such that f(n) >cg(n) for all n, n ≥ n0 that is, f has a higher growth rate than g so little omega (ω) is to
mean “tight lower bound”. The main difference between Big Omega (Ω) and little omega (ω) lies in their
definitions. In the case of Big Omega f(n) = Ω(g(n)) and the bound is 0<= cg(n)<f(n) so it is true only for
some positive value of ‘c>0’ but in case of little omega; it is true for all constant c>0. The little oh (o) can
also be defined as a functions of limits as given below

 )(
)(
)( lim ng
nf
iffog
n
Example prove that 4n+6=ω(1)
Solution
The Little omega (ω) running time can be proved by applying the limit formula as given below
If
)(
)(
lim ng
nf
n 
=∞, then functions f(n) is ω(g(n))
Here we have
(1))(and64n)(  ngnf
So 



)
64n
(
1
64n
)(
)(
limlimlim nnn ng
nf
Therefore f(n) is ω(g(n)) so
4n+6=ω(1)
0
)f(
)g(
)( lim 
 n
n
iffog
n