# Essential Big O Notations Every Developer Must Master

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## Chapter 1: Introduction to Big O Notation

When preparing for coding interviews, a fundamental concept to grasp is Big O notation. In the realms of computer science and software development, efficiency is paramount. Whether optimizing code, crafting algorithms, or designing systems, it is essential to comprehend the performance characteristics of your algorithms.

Big O notation serves as a standardized method to articulate both the time and space complexity of algorithms, allowing developers to quantitatively analyze and compare various approaches. In most coding interviews, interviewers will inquire about the time and space complexity of your proposed solutions, often prompting discussions on potential optimizations—this is where the trade-off between time and space comes into play.

Previously, I’ve covered several system design interview topics, including the differences between API Gateways and load balancers, as well as Forward Proxy versus Reverse Proxy. In this article, we will delve into eight fundamental Big O notations that every developer should be familiar with to create efficient and scalable code. If you're pressed for time, here’s a quick summary of essential Big O notations, ranked from best to worst:

### 1. O(1) — Constant Time Complexity

Algorithms exhibiting O(1) time complexity have a consistent execution time, which remains unchanged regardless of the input size. This is the most favorable performance for any algorithm. For instance, accessing an element in an array using its index, performing basic arithmetic operations, or retrieving a property of an object all qualify as constant time O(1) operations. The performance remains steady whether your array contains one element or one billion.

Here is a visual representation of O(1) performance:

### 2. O(log n) — Logarithmic Time Complexity

Algorithms with O(log n) time complexity exhibit a runtime that increases logarithmically as input size grows. This is the second most favorable performance after O(1). If optimizing for O(1) isn’t feasible, aim for O(log n) for your algorithms. A prime example is the binary search algorithm, which repeatedly halves the input space.

Here’s what logarithmic complexity looks like:

### 3. O(n) — Linear Time Complexity

Algorithms that demonstrate O(n) time complexity have a runtime that scales linearly with input size. This implies that as the amount of data increases, the performance of your algorithms declines proportionately. Many algorithms fall into this category, such as linear search and various linked list operations. Iterating through an array or list to perform an operation on each element exemplifies linear time complexity.

Here’s a graph depicting linear time complexity:

### 4. O(n log n) — Linearithmic Time Complexity

Algorithms with O(n log n) time complexity have a runtime that increases in proportion to n times the logarithm of n, making them less efficient than linear algorithms due to the additional log factor. Efficient sorting algorithms like merge sort, quicksort, and heapsort fit into this category.

Here’s a graph illustrating O(n log n) performance:

### 5. O(n²) — Quadratic Time Complexity

Algorithms with O(n²) time complexity exhibit a runtime that grows quadratically with the input size. These algorithms tend to be slow, and if you present a solution with quadratic time complexity, interviewers will likely encourage you to optimize it down to an acceptable O(n) or O(log n) level. Examples include nested loops where each iteration performs a linear operation, such as bubble sort or selection sort.

Here’s a representation of quadratic time complexity:

### 6. O(2^n) — Exponential Time Complexity

Algorithms with O(2^n) time complexity are among the slowest and are typically undesirable. Their runtime doubles with each added input element. You can enhance performance of such algorithms through caching and memoization to avoid redundant calculations. A notable example is the naive recursive solution for the Fibonacci sequence.

Here’s how exponential time complexity looks:

### 7. O(n!) — Factorial Time Complexity

Algorithms characterized by O(n!) time complexity have a runtime that increases factorially with input size. These algorithms are also regarded as slow and usually undesirable. If your solution has factorial time complexity, be prepared for interviewers to seek improvements, as this complexity is rarely acceptable unless the problem is exceptionally complex. An example is brute force algorithms that generate all permutations or combinations of a set.

Here’s a visual of factorial time complexity:

### 8. O(n^c) — Polynomial Time Complexity

Algorithms with O(n^c) time complexity have a runtime that grows polynomially with the input size, where c is a constant. This represents the slowest type of algorithm and is the last in our essential Big O notation list. Matrix multiplication algorithms, such as the Strassen algorithm, are examples of polynomial time complexity.

Here’s a graph that illustrates the various Big O notations concerning time, highlighting how performance deteriorates from O(1) to O(2^n):

## Conclusion

This concludes our overview of the eight essential Big O notations every developer should familiarize themselves with. Mastery of Big O notation empowers developers to objectively assess the efficiency and scalability of their algorithms. By understanding and internalizing these key Big O notations, you can make informed decisions in algorithm selection or design, optimize performance, and enhance the scalability of your software solutions. Continuously improving your skills in algorithm analysis and complexity theory will undoubtedly bolster your ability to write efficient and robust code across diverse areas of software development.

The first video, "The Complete Guide to Big O Notation & Complexity Analysis for Algorithms: Part 1 of 2," provides an in-depth examination of Big O notation, focusing on its significance in algorithm efficiency and performance analysis.

The second video, "What is Big O Notation, and Why You Should Care," explains the fundamentals of Big O notation and its importance in software development and coding interviews.