What is asymptotic notation? Describe different types of asymptotic notation with example.
Asymptotic Notation: Asymptotic
notation are used to describe the asymptotic running time of an algorithm that
are defined in terms of functions whose domains are the set of natural numbers N = { 0, 1, 2, ----- }.
Such notations are convenience for
describing the worst case running time function T(n), Which is usually defined only on an integer
input size.
Asymptotic notation are three types:
These are_
(i)
Big-O Notation.
(ii)
Omega- Ω notation
(iii)
Theta- 𑁜 notation
(i)
Big- O Notation:- The
function f(n)= O(g(n)), if there exist positive constants C and n0
such that f(n) ≤ c* g(n) for all n, n ≥ n0.
We used O-notation to give an upper bound on a function to within a constant
factor.
fig-1 shows the intuition behind big O- notation for all
values n to the right of n0, the value of the function f(n) is on or
below g(n).
Example of O:
3n+2= O(n), since 3n+2 ≤ 4n, for all n ≥ 2.
(ii) Ω (Omega) Notation:
The function f(n) = Ω (g(n)) if there exist positive
constants c and n0, such that f(n) ≥ c * g(n) for all n, n ≥ n0.
We use Ω notation to give a lower bound on a function within a constant factor.
fig-1 shows the intuition behind Ω notation for all values n to
the right of n0, the value of the function f(n) is on or above g(n).
Example of Ω:
3n+2= Ω(n), since
3n+2n ≥ 3n, for all n ≥ 1.
(iii) 𑁜(Theta)
Notation: The function f(n) = 𑁜 (g(n)) if there exists positive constants C1,
C2, C3 and n0 such that 0≤ C1 g(n)
≤ f(n) ≤ C2 g(n), We use 𑁜 notation to give a tight bound on a function
within a constant factor.
Fig-1 shows the in intuition behind 𑁜 notation. For all values n to the right of n0,
the value of the function f(n) is at above c,g(n) and at or below c2g(n).
Example of 𑁜 :
If f(n)= 3n+6 and g(n), then f(n) = 𑁜(g(n), or 3n+6=𑁜(n), as can be seen by
using C1=3, C2=4 and N=5.
No comments
Dear Members, Thanks for Your Comments. We must be reply your comment answer as soon as possible. Please Stay with us.....