TIME COMPLEXITY A LEVEL

What is time complexity

Time complexity is a measure of how the runtime of an algorithm increases with respect to the input size (n). It helps us evaluate the efficiency of an algorithm.

Common Types of Time Complexity:

Constant Time – O(1)

• The execution time remains the same regardless of input size.

• Example: Accessing an element in an array by index → arr[5]

• The runtime grows slowly as the input size increases.

• Example: Binary search (dividing the problem in half each step).

Linear Time – O(n

• The runtime increases proportionally with the input size.

• Example: Iterating through an array → for i in range(n)

Logarithmic Time – O(log n)

• The runtime grows proportionally to the square of the input size.

• Example: Nested loops iterating through an array → for i in range(n): for j in range(n):

Quadratic Time – O(n^2)

• The runtime is a combination of linear and logarithmic growth.

• Example: Efficient sorting algorithms like Merge Sort and Quick Sort (average case).

Linearithmic Time – O(nlog⁡n)

• The runtime grows slowly as the input size increases.

• Example: Binary search (dividing the problem in half each step).

Linear Time – O(n

• The runtime increases proportionally with the input size.

• Example: Iterating through an array → for i in range(n)

Logarithmic Time – O(log n)

• The runtime grows extremely fast.

• Example: Generating all possible permutations of n elements

Factorial Time – O(n!)

• The runtime increases proportionally with the input size.

• Example: Iterating through an array → for i in range(n)

Exponential Time – O(2^n)

​Big O Notation

• Best Case: The minimum time required for execution.

• Average Case: The expected time taken for random input.

• Worst Case: The maximum time required (important for analysis).