Big Oh Notation, Big Omega Notation, Big Theta Notation

1. Asymptotic Notations

2. Introduction
 In mathematics, computer science, and related fields, big O
notation describes the limiting behavior of a function when the
argument tends towards a particular value or infinity, usually in
terms of simpler functions.
 The time complexity of an algorithm quantifies the amount of
time taken by an algorithm to run as a function of the size of the
input to the problem. When it is expressed using big O notation,
the time complexity is said to be described asymptotically, i.e.,
as the input size goes to infinity.

3. 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(n2).
 Describes behavior of function in the limit.
 Written using Asymptotic Notation.
 The notations describe different rate-of-growth relations between
the defining function and the defined set of functions.

4. Big O-notation
O(g(n)) = {f(n) :
 positive constants c and n0, such
that n  n0,
we have f(n)  cg(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).
Example:
i) f(n)=3n+2 and g(n)=n Prove f(n)  cg(n)
ii) f(n)=100n+5 and g(n)=n2 Prove f(n)  cg(n)

5. f(n)=n^2 g(n)=2^n
Big O Notation

6.  -notation
g(n) is an asymptotic lower bound
for f(n).
Example:
i) f(n)=3n+2 and g(n)=n Prove f(n) >= cg(n)
ii) f(n)=n3 and g(n)=n2 Prove f(n) >= cg(n)
(g(n)) = {f(n) :
 positive constants c and n0, such
that n  n0,
we have cg(n)  f(n) or
f(n)>=cg(n)}
For function g(n), we define (g(n)), big-
Omega of n, as the set:

7. Big Omega Notation

8. Q-notation
Q(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
such that n  n0,
we have c1g(n)  f(n)  c2g(n)
}
For function g(n), we define Q(g(n)), big-
Theta of n, as the set:
g(n) is an asymptotically tight bound
for f(n).
Example:
i) f(n)=3n+2 and g(n)=n Prove

9. Relations Between Q, O, 

10. Basic Asymptotic Efficiency classes
C constant, we write
O(1)
logN logarithmic
log2N log-squared
N linear
NlogN
N2 quadratic
N3 cubic
2N exponential
N! factorial

11. Properties of Asymptotic Notation
1. Transitive
If f(n) = Θ(g(n)) and g(n) = Θ(h(n)), then f(n) = Θ(h(n))
If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n))
If f(n) = Ω(g(n)) and g(n) = Ω(h(n)), then f(n) = Ω(h(n))
2. Reflexivity
f(n) = Θ(f(n))
f(n) = O(f(n))
f(n) = Ω(f(n))
3. Symmetry
f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))
4. Transpose Symmetry
f(n) = O(g(n)) if and only if g(n) = Ω(f(n))

12. The End