What exactly does big Ө notation represent?

Question

I\'m really confused about the differences between big O, big Omega, and big Theta notation.

I understand that big O is the upper bound and big Omega is the lower b

Answer 1

First of All Theory

1. Big O = Upper Limit O(n)

2. Theta = Order Function - theta(n)

3. Omega = Q-Notation(Lower Limit) Q(n)

Why People Are so Confused?

In many Blogs & Books How this Statement is emphasised is Like

"This is Big O(n^3)" etc.

and people often Confuse like weather

O(n) == theta(n) == Q(n)

But What Worth keeping in mind is They Are Just Mathematical Function With Names O, Theta & Omega

so they have same General Formula of Polynomial,

Let,

f(n) = 2n4 + 100n2 + 10n + 50 then,

g(n) = n4, So g(n) is Function which Take function as Input and returns Variable with Biggerst Power,

Same f(n) & g(n) for Below all explainations

Big O - Function (Provides Upper Bound)

Big O(n4) = 3n4, Because 3n4 > 2n4

3n4 is value of Big O(n4) Just like f(x) = 3x

n4 is playing a role of x here so,

Replacing n4 with x'so, Big O(x') = 2x', Now we both are happy General Concept is

So 0 ≤ f(n) ≤ O(x')

O(x') = cg(n) = 3n4

Putting Value,

0 ≤ 2n4 + 100n2 + 10n + 50 ≤ 3n4

3n4 is our Upper Bound

Theta(n) Provides Lower Bound

Theta(n4) = cg(n) = 2n4 Because 2n4 ≤ Our Example f(n)

2n4 is Value of Theta(n4)

so, 0 ≤ cg(n) ≤ f(n)

0 ≤ 2n4 ≤ 2n4 + 100n2 + 10n + 50

2n4 is our Lower Bound

Omega n - Order Function

This is Calculated to find out that weather lower Bound is similar to Upper bound,

Case 1). Upper Bound is Similar to Lower Bound

if Upper Bound is Similar to Lower Bound, The Average Case is Similar

Example, 2n4 ≤ f(x) ≤ 2n4,
Then Omega(n) = 2n4

Case 2). if Upper Bound is not Similar to Lower Bound

in this case, Omega(n) is Not fixed but Omega(n) is the set of functions with the same order of growth as g(n).

Example 2n4 ≤ f(x) ≤ 3n4, This is Our Default Case,
Then, Omega(n) = c'n4, is a set of functions with 2 ≤ c' ≤ 3

Hope This Explained!!