Asymptotic Notation

Asymptotic Notation

Asymptotic notation is a mathematical notation used to describe the limiting behavior of a function as its input approaches infinity. It is commonly employed in computer science and mathematics to analyze the efficiency and performance of algorithms.

Big O Notation ($O$):

• Definition: $f(n)$ is $O(g(n))$ if there exist positive constants $c$ and $n_0$ such that $0 \leq f(n) \leq c \cdot g(n)$ for all $n \geq n_0$.
• Explanation: Big O notation provides an upper bound on the growth rate of a function. It represents the worst-case scenario in terms of time or space complexity.
• Example: $f(n) = 2n^2 + 3n + 1$ is $O(n^2)$ because $2n^2 + 3n + 1 \leq c \cdot n^2$ for some constant $c$ when $n$ is sufficiently large.

Omega Notation ($\Omega$):

• Definition: $f(n)$ is $\Omega(g(n))$ if there exist positive constants $c$ and $n_0$ such that $0 \leq c \cdot g(n) \leq f(n)$ for all $n \geq n_0$.
• Explanation: Omega notation provides a lower bound on the growth rate of a function. It represents the best-case scenario in terms of time or space complexity.
• Example: $f(n) = 2n^2 + 3n + 1$ is $\Omega(n^2)$ because $c \cdot n^2 \leq 2n^2 + 3n + 1$ for some constant $c$ when $n$ is sufficiently large.

Theta Notation ($\Theta$):

• Definition: $f(n)$ is $\Theta(g(n))$ if it is both $O(g(n))$ and $\Omega(g(n))$.
• Explanation: Theta notation provides a tight bound on the growth rate of a function, indicating both upper and lower bounds.
• Example: $f(n) = 2n^2 + 3n + 1$ is $\Theta(n^2)$ because it is both $O(n^2)$ and $\Omega(n^2)$.

অ্যালগরিদম কমপ্লেক্সিটি(বিগ “O” নোটেশন)