Asymptotic Notation

Asymptotic notation is a way of expressing the cost of an algorithm. Goal of Asymptotic notation is to simplify Analysis by getting rid of unneeded information

Following are the asymptotic notation:

Big–Oh Notation (O) :
>> O(g(n)) = { f(n) : there exist positive constants c and n0 such that 0 < f(n) < c*g(n) for all n > n0 }.

>> It is asymptotic upper bound.

>>The function f(n) = O(g(n)) iff there exist positive constants c and n0 such that f(n) ≤ c * g(n) for all n, n ≥ n0.

>> The statement f(n) = O(g(n)) states only that g(n) is an upper bound on the value of f(n) for all n, n ≥ no.

>> Eg :
1. 3n + 2 = O(n)
3n + 2 ≤ 4n for all n ≥ 2.
2. 3n + 3 = O(n)
3n + 3 ≤ 4n for all n ≥ 3.
3. 100n + 6 = O(n)
100n + 6 ≤ 101n for all n ≥ 6.
Omega Notation (Ω)
>> Ω (g (n)) = { f(n) : there exist positive constants c and n0 such that 0 < c * g(n) < f(n) for all n > n0 }

>> Asymptotic lower bound.

>> The function f(n) = Ω(g(n)) iff there exist positive constants c and n0 such that f(n) ≥ c * g(n) for all n, n ≥ n0.

>> The statement f(n) = Ω(g(n)) states only that g(n) is only a lower bound on the value of f(n) for all n, n ≥ no.

>> E.g.
1. 3n + 2 = Ω(n)
3n + 2 ≥ 3n for all n ≥ 1.
2. 3n + 3 = Ω(n)
3n + 3 ≥ 3n for all n ≥ 1.
3. 100n + 6 = Ω(n)
100n + 6 ≥ 100n for all n ≥ 0.
Theta Notation (Θ)
>> Θ(g(n)) = { f(n) : there exist positive constants c1 and c2 and n0 such that 0 < c1 * g(n) < f(n) < c2  * g(n) for all n > n0 }

>> The function f(n) = Θ(g(n)) iff there exist positive constants C1, C2 and n0 such that C1 * g(n) ≤ f(n) ≤ C2 * g(n) for all n, n ≥ n0.

>> E.g.
1. 3n + 2 = Θ(n)
3n + 2 ≥ 3n for all n ≥ 2.
3n + 2 ≤ 4n for all n ≥ 2.
So, C1 = 3, C2 =4 & n0 =2.

>> The statement f(n) = Θ(g(n)) iff g(n) is both an upper and lower bound on the value of f(n).

Little oh Notation (o)
>> o(g(n)) = { f(n) : for any positive constant c>0 , there exists a constant n0 such that 0 < f(n) < c  * g(n) for all n > n0 }

>> We use o notation to denote an upper bound that is not asymptotically tight.

>> The definitions of O-notation and o-notation are similar. The main difference is that in f(n) =O(g(n)), the bound 0 < f(n) < c*g(n) holds for some constant c>0 but in f(n) =O(g(n)), the bound 0 < f(n) < c *g(n) holds for all constants c > 0.
Little omega Notation (w)
>> w(g(n)) = { f(n) : for any positive constant c>0 , there exists a constant n0 such that 0 < c * g (n) < f(n) for all n > n0 }


>> We use w notation to denote an lower bound that is not asymptotically tight.