Define asymptotic notation

Asymptotic notation is a mathematical tool used in computer science to describe the behavior of algorithms in terms of their growth rates as the input size approaches infinity. It helps to classify algorithms based on their efficiency and scalability. Here are the main types of asymptotic notation:

1. Big O Notation (O)

• Definition: Big O notation describes an upper bound on the time or space complexity of an algorithm. It provides a worst-case scenario of how the algorithm’s running time or space requirements grow with the size of the input.
• Expression: If $f(n) = O(g(n))$, it means there exist constants $C$ and $n_0$ such that for all $n \geq n_0$, $f(n) \leq C \cdot g(n)$.
• Example: For a linear search algorithm, the time complexity is $O(n)$, indicating that the running time grows linearly with the input size.

2. Big Omega Notation (Ω)

• Definition: Big Omega notation describes a lower bound on the time or space complexity of an algorithm. It represents the best-case scenario or a minimum growth rate.
• Expression: If $f(n) = \Omega(g(n))$, it means there exist constants $C$ and $n_0$ such that for all $n \geq n_0$, $f(n) \geq C \cdot g(n)$.
• Example: A linear search algorithm has a best-case time complexity of $\Omega(1)$ when the target element is found at the first position.

3. Big Theta Notation (Θ)

• Definition: Big Theta notation provides a tight bound on the time or space complexity of an algorithm. It describes both an upper and a lower bound, giving a precise asymptotic behavior.
• Expression: If $f(n) = \Theta(g(n))$, it means there exist constants $C_1$, $C_2$, and $n_0$ such that for all $n \geq n_0$, $C_1 \cdot g(n) \leq f(n) \leq C_2 \cdot g(n)$.
• Example: For a merge sort algorithm, the time complexity is $\Theta(n \log n)$, indicating that the running time grows as $n \log n$ in both the best and worst cases.

4. Little o Notation (o)

• Definition: Little o notation describes an upper bound that is not tight. It indicates that the function grows strictly slower than the comparison function.
• Expression: If $f(n) = o(g(n))$, it means for every positive constant $\epsilon$, there exists a constant $n_0$ such that for all $n \geq n_0$, $f(n) < \epsilon \cdot g(n)$.
• Example: If $f(n) = o(n^2)$, it means that $f(n)$ grows slower than $n^2$.

5. Little omega Notation (ω)

• Definition: Little omega notation describes a lower bound that is not tight. It indicates that the function grows strictly faster than the comparison function.
• Expression: If $f(n) = \omega(g(n))$, it means for every positive constant $\epsilon$, there exists a constant $n_0$ such that for all $n \geq n_0$, $f(n) > \epsilon \cdot g(n)$.
• Example: If $f(n) = \omega(n)$, it means that $f(n)$ grows faster than $n$.