Big-Omega Notation (Ω-notation) is used to describe the lower bound of an algorithm's growth rate. It provides a way to express the best-case scenario for how an algorithm performs as the input size increases.

• Purpose: It tells us the minimum amount of time or space required by an algorithm, even in the best-case scenario.
• Usefulness: While Big-O notation gives an upper bound (worst-case), Ω-notation helps understand the least amount of resources an algorithm will always need.

Formal Definition of Big-Omega Notation

A function f(n) is said to be \Omega(g(n)) if there exist positive constants c and n_0 such that:

f(n) \geq c \cdot g(n)

for all n \geq n_0.

• c is a positive constant that scales the function g(n).
• n_0 is a threshold value such that the inequality holds for all larger n.

Understanding Big-Omega Through an Example

Let's find the \Omega notation for a function:

Example: f(n) = 4n^2 + 3n + 6

1. Identify the Dominant Term:

   The dominant term here is 4n^2, which grows faster than the terms 3n and 6 when n is large.

2. Choose g(n) as the Dominant Term:

   Let’s set g(n) = n^2. We need to find a constant c such that:

4n^2 + 3n + 6 \geq c \cdot n^2

3. Simplify the Inequality:

   To find a suitable constant c, we can observe that as n becomes large, the terms 3n and 6 contribute less to the growth compared to 4n^2. Therefore, we can choose:

c = 2

This choice satisfies the inequality for all n \geq 1.

4. Conclusion:

   Since we found a constant c and a threshold n_0 = 1, we conclude that:

f(n) = \Omega(n^2)

How Does Big-Omega Compare with Big-O?

• Big-Omega (Ω-notation): Describes the lower bound, giving us a sense of the minimum growth rate of an algorithm.
• Big-O Notation: Represents the upper bound, describing the maximum growth rate.

Properties of Big-Omega Notation

1. Lower Bound Description: Ω-notation provides the minimum number of steps required for an algorithm.
2. Best-Case Scenario: It’s used when we want to analyze the most favorable outcome.
3. Tight Bounds with Big-Theta: If an algorithm's growth rate is tightly bounded both above and below, then f(n) can be expressed using Θ-notation.