Time Complexity

1. Overview

Definition of Time Complexity: A computational complexity that describes the amount of time it takes to run an algorithm as a function of the length of the input.
Big O Notation: Provides an upper bound on the time complexity, allowing for worst-case scenario analysis. Common notations include:
- O(1): Constant time
- O(log n): Logarithmic time
- O(n): Linear time
- O(n log n): Linearithmic time
- O(n²): Quadratic time
- O(2ⁿ): Exponential time
Complexity Classes:
- P: Problems that can be solved in polynomial time.
- NP: Problems for which solutions can be verified in polynomial time.
- NP-complete: Hardest problems in NP, which if solved in polynomial time would imply P = NP.

What specific algorithm or data structure are you focusing on in the context of time complexity?
Are there particular properties of time complexity (e.g., average case vs. worst case) you wish to explore further?

How do different data structures (arrays, linked lists, trees) impact time complexity?
In what scenarios do average case complexities differ significantly from worst-case analyses?
What are the implications of time complexity in real-time systems versus batch processing systems?

Upper Bound: This is represented as Big O notation (e.g., O(n)), indicating the maximum amount of time an algorithm may take to complete.
Lower Bound: Represented as Big Omega notation (e.g., Ω(n)), which indicates the minimum time necessary for a given algorithm for every input size.
Tight Bound: Denoted as Big Theta notation (e.g., Θ(n)), which describes both the upper and lower bounds of an algorithm, providing an accurate characterization of performance.
Amortized Complexity: Average case over a sequence of operations. It is useful for data structures that may have occasional expensive operations (e.g., dynamic array resizing).
Recursive Algorithms: Time complexities for recursive algorithms often derive from the Master Theorem or recurrence relations to determine their performance characteristics.