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Prerequisite – Asymptotic Notations, Properties of Asymptotic Notations
1. Big Oh notation (O) :
Big oh notation is used to describe asymptotic upper bound.
Mathematically, if f(n) describe running time of an algorithm; f(n) is O(g(n)) if there exist positive constant C and no such that
0 <=f(n) <= c g(n) for all n>=n0
n = used to give upper bound an a function.
If a function is O(n), it is automatically O(n-square) as well !
Graphic example for Big oh (O) :
2. Big Omega notation (Ω) :
Just like O notation provide an asymptotic upper bound, ? notation provides asymptotic lower bound.
Let f(n) define running time of an algorithm;
f(n) is said to be Ω(g (n)) if there exists positive constant C and (n 0 ) such that
O<= C g(n) <= f(n) for all n>=n 0
n= used to given lower bound on a function
If a function is O(n-square ) it is automatically O(n) as well.
Graphical example for Big Omega (Ω) :
3. Big Theta notation (Θ) :
Let f(n) define running time of an algorithm.
f(n) is said to be Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n))
Mathematically,
O<=f(n)<=C 1 g(n) for n>=n 0 O<= C 2 g(n)<=f(n) for n >=n 0
Merging both the equation, we get :
O<=C 2 g(n)<=f(n)<=C 1 g(n) for n>=n 0
The equation simply means there exist positive constants C 1 and C 2 such that f(n) is sandwich between C 2 g(n) and C 1 g(n).
Graphic example of Big Theta (Θ) :
Difference Between Big oh, Big Omega and Big Theta :
| S.No. | Big Oh | Big Omega | Big Theta |
|---|---|---|---|
| 1. |
It is like <= rate of growth of an algorithm is less than or equal to a specific value |
It is like >= rate of growth is greater than or equal to a specified value |
It is like == meaning the rate of growth is equal to a specified value |
| 2. | The upper bound of algorithm is represented by Big O notation. Only the above function is bounded by Big O. asymptotic upper bond is it given by Big O notation. | The algorithm’s lower bound is represented by Omega notation. The asymptotic lower bond is given by Omega notation | The bonding of function from above and below is represented by theta notation. The exact asymptotic behavior is done by this theta notation. |
| 3. | Big oh (O) – Worst case | Big Omega (Ω) – Best case | Big Theta (Θ) – Average case |
| 4. | Big-O is a measure of the longest amount of time it could possibly take for the algorithm to complete. | Big- Ω is take a small amount of time as compare to Big-O it could possibly take for the algorithm to complete. | Big- Θ is take very short amount of time as compare to Big-O and Big-? it could possibly take for the algorithm to complete. |
| 5. | Mathematically – Big Oh is 0 <=f(n) <= c g(n) for all n>=n0 | Mathematically – Big Omega is O<= C g(n) <= f(n) for all n>=n 0 | Mathematically – Big Theta is O<=C 2 g(n)<=f(n)<=C 1 g(n) for n>=n 0 |
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