Algorithmic Analysis | Small o and small Omega symbols

Asymptotic Analysis the main idea is to measure the efficiency of an algorithm that does not depend on machine-specific constants, mainly because the analysis does not require algorithms to be implemented and to compare the time spent by the program. We have discussed three major asymptotic symbols . The time complexity of the algorithm is expressed using the following 2 asymptotic symbols.

Small ο Asymptotic symbol

large o is used as a tight upper limit of the growth of the algorithm effort (this effort is described by the function f (n)), although as mentioned above it can also be loosely capped. the "ο" (ο ()) notation is used to describe an upper limit that cannot be tightened.

definition: orders f (n) and g (n) are functions that map positive integers to positive real numbers. if for any real constant c> 0, the F (n) is ο (g (n)) (or f (n) ∈ (g (n)) , there is an integer constant n0≥1, which makes F (n) 0.  It means that small O () represents the f (n) loosely bound .   In mathematical relationships,  f (n) = O (g (n)) means  lim f (n)/g (n) = 0  n→∞

For example:

7n + 8∈o (n 2 )?
To do so, for any C, we must be able to find the
f (N) <c * g (n) asymptotically true N0 .
Let's give an example,
if C = 100, we check whether inequality is clear. If C = 1/100, we will have to use
more imagination, but we can find a n0. (Try n0 = 1000.) from
In these examples, the speculation seems to be correct.
Then check the limits,
Lim F (n)/g (n) = Lim (7n + 8)/(n 2 ) = Lim 7/2n = 0 (l ' Hospital)
n→∞ n→∞ n→∞

so 7n + 8∈o (n 2 )

Small Ω-Asymptotic symbol

Definition: so that f (n) and g (n) are functions that map positive integers to positive real numbers. we say f (n) is Ω (g (n)) (or f (n) ∈ω (g (n))) if for any real constant c> 0, there is an integer constant n0≥1, (n) > C * g (n) ≥0, for integer n≥n0.

F (n) has a higher growth rate than g (n), so the main difference between large omega (Ω) and small omega (ω) is their definition. In large Omega F (n) =ω (g (n)), and the boundary is 0 <= CG (n) 0, but in the case of small Ω, for all constants c> 0 is so.

We use the Ω notation to denote a lower bound that is not asymptotically tight.
F (N) ∈ω (g (n)) if and only if G (n) ∈ο ((f (n))),

In the mathematical relationship,
If f (n) ∈ω,

Lim f (N)/g (n) =∞

n→∞ Example:

Proof 4n + 6∈ο (1);
The Ω (ο) run time can be demonstrated by applying the restriction formula given below.
If Lim f (n)/g (n) =∞, then the function f (n) is ο (g (n))
n→∞
Here we have the function f (n) = 4n + 6 and g (n) = 1
Lim (4n + 6)/(1) =∞
n→∞ For any C, we can get n0 for this inequality 0 <= c * g (n) <f (n), 0 <= c * 4n + 6
So prove.

Algorithmic Analysis | Small o and small Omega symbols