Computational Complexity

TL;DR

• Measures how difficult a problem or task is for a computer by quantifying time, space, and other resources.
• Expressed as growth rates (e.g., $O(n^2)$, $O(nlogn)$, $O(n!)$) that indicate how resource needs scale with input size.
• Guides selection of algorithms and assessment of feasibility and efficiency.

Definition

Computational complexity refers to the amount of time, space, and resources required to solve a given problem or perform a specific task.

Explanation

Computational complexity is a measure of how difficult it is to solve a problem or perform a task using a computer. It quantifies the time, space, and other resources an algorithm or computational process requires, typically expressed as how those requirements grow with the size of the input. Understanding computational complexity helps determine the feasibility and efficiency of algorithms and computational tasks.

Examples

Sorting algorithms

Different sorting algorithms have varying levels of computational complexity. For instance, the bubble sort algorithm has a time complexity of $O(n^2)$, meaning that it takes longer to sort a large number of items compared to other sorting algorithms with a lower time complexity, such as the quicksort algorithm, which has a time complexity of $O(nlogn)$.

Traveling salesperson problem

The traveling salesperson problem involves finding the shortest possible route that a salesperson can take to visit a given set of cities and return to the starting point. The time complexity of this problem is $O(n!)$, meaning that it becomes increasingly difficult to solve as the number of cities increases.

Use cases

• Determining the feasibility and efficiency of algorithms and other computational tasks.
• Selecting the most appropriate methods for solving specific problems and tasks.
• Identifying ways to improve the efficiency of existing algorithms.

Notes or pitfalls

• Problems with factorial or similarly rapid growth in time complexity (for example, $O(n!)$) become increasingly difficult to solve as input size increases.

Related terms

• Sorting algorithms
• Bubble sort
• Quicksort
• Traveling salesperson problem