Asymptotic analysis is a method for describing the efficiency of algorithms by analyzing their performance as the input size grows towards infinity. Rather than focusing on the exact number of operations, it characterizes algorithm performance in terms of how the resource usage (time or space) scales with input size.

Key principles of asymptotic analysis include:

• Focus on growth rate: Examine how an algorithm's efficiency changes as input size increases
• Ignore constants: Constant factors don't affect the overall growth rate (e.g., 2n and 100n both grow linearly)
• Ignore lower-order terms: As input size increases, higher-order terms dominate (e.g., n² + n becomes approximately n² for large n)

Asymptotic analysis typically uses three main notations:

• Big O (O): Upper bound - the function grows no faster than
• Big Omega (Ω): Lower bound - the function grows at least as fast as
• Big Theta (Θ): Tight bound - the function grows at exactly the same rate as

This approach allows us to compare algorithms independently of implementation details, hardware, or specific inputs, focusing instead on their fundamental efficiency characteristics.