What is Algorithm Complexity?

Algorithm complexity refers to the computational resources required by an algorithm to solve a problem. These resources are typically measured in terms of time (time complexity) and space (space complexity). By analyzing algorithm complexity, you can determine how well your code will perform, especially as the size of the input data grows.

Notations

Theta represents the best case complexity.

Omega represents the average case complexity.

Big O represents the worst case complexity.

Usually in interviews, you only need the Big O complexity.

Types

O(1): Constant Time Complexity

An algorithm with O(1) complexity takes the same amount of time to execute regardless of the input size. This is the most efficient time complexity.

O(log n): Logarithmic Time Complexity

Algorithms with O(log n) complexity reduce the problem size with each step, such as binary search. They are highly efficient, especially for large datasets.

O(n): Linear Time Complexity

An algorithm with O(n) complexity scales linearly with the input size. Common examples include simple loops and iterations.

O(n log n): Linearithmic Time Complexity

This complexity is common in efficient sorting algorithms like mergesort and heapsort. It combines linear and logarithmic complexities.

O(n^2): Quadratic Time Complexity

Algorithms with O(n^2) complexity involve nested loops, resulting in performance that scales quadratically with the input size. Examples include selection sort and bubble sort.

O(2^n): Exponential Time Complexity

These algorithms have a growth rate that doubles with each additional input element, such as certain recursive algorithms. They are impractical for large inputs.

O(n!): Factorial Time Complexity

The most inefficient algorithms, with performance that scales factorially with input size. These are rare and usually signify a brute-force approach.

Source: Big O Cheat Sheet

Analyzing Algorithm Complexity

When analyzing the complexity of an algorithm, consider the following steps:

Identify the Basic Operations

Determine the fundamental operations that dominate the algorithm's performance, such as comparisons, assignments, or arithmetic operations.

Count the Operations

Analyze the algorithm's structure to count how many times these basic operations are performed concerning the input size.

Express in Big O Notation

Simplify the expression by ignoring constants and lower-order terms, focusing on the dominant term that describes the growth rate.

Optimizing Algorithm Complexity

Here are some tips to optimize the complexity of your algorithms:

Choose Efficient Data Structures

The choice of data structures can significantly impact algorithm performance. For example, using a hash table instead of a list can reduce lookup times from O(n) to O(1).

Avoid Unnecessary Computations

Eliminate redundant calculations and optimize loops to reduce the number of operations.

Divide and Conquer

Use algorithms that break the problem into smaller subproblems, solve them independently, and combine the results. This approach can often lead to more efficient solutions.

Consider Space Complexity

While optimizing time complexity, don't overlook space complexity. Efficient memory usage can also enhance performance, especially in memory-constrained environments.

Data Structures Operations

Source: Big O Cheat Sheet

Heap Operations

Source: Big O Cheat Sheet

Array Sorting Algorithms

Source: Big O Cheat Sheet

Searching Algorithms

Source: Big O Cheat Sheet

Graph Algorithms

Source: Big O Cheat Sheet