1. What Is Time Complexity?

Time complexity is a computational concept that describes the amount of time an algorithm takes to run as a function of the length of the input. It's expressed using Big O notation, which provides an upper bound on the growth rate of the running time.

2. How To Calculate Time Complexity

The fundamental formula for time complexity is:

Where:

• \( T(n) \) — Time complexity as a function of input size
• \( O \) — Big O notation (asymptotic upper bound)
• \( f(n) \) — Function describing growth rate
• \( n \) — Input size

Explanation: Time complexity analysis focuses on how the runtime grows as the input size increases, ignoring constant factors and lower-order terms.

3. Common Time Complexities

O(1) - Constant Time: Runtime doesn't depend on input size (e.g., array access)
O(log n) - Logarithmic Time: Runtime grows logarithmically (e.g., binary search)
O(n) - Linear Time: Runtime grows linearly with input size (e.g., linear search)
O(n log n) - Linearithmic Time: Common in efficient sorting algorithms (e.g., merge sort)
O(n²) - Quadratic Time: Runtime grows with square of input size (e.g., bubble sort)
O(2ⁿ) - Exponential Time: Runtime doubles with each additional input (e.g., brute force algorithms)

4. Using The Calculator

Tips: Select the algorithm type, enter input size (n), and provide base operations. The calculator will estimate total operations and display the Big O notation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between time and space complexity?
A: Time complexity measures runtime growth, while space complexity measures memory usage growth as input size increases.

Q2: Why do we ignore constants in Big O notation?
A: Constants become insignificant for large inputs, and Big O focuses on asymptotic behavior and growth rates.

Q3: What is the best time complexity?
A: O(1) is ideal, but O(log n) and O(n) are also considered efficient for most practical purposes.

Q4: How do I analyze time complexity of nested loops?
A: Multiply the complexities of each loop. Two nested O(n) loops become O(n²), three become O(n³).

Q5: When is exponential complexity acceptable?
A: Only for very small input sizes, typically n ≤ 20-30, as it becomes infeasible for larger inputs.