# Introduction to Big O Notation

## Understanding Algorithm Complexity

Big O notation is a mathematical tool that allows us to express the time and space complexity of algorithms in a clear and concise way. It’s an essential concept in computer science, as it helps us analyze the efficiency of algorithms and make informed decisions about which ones to use in different situations.

## What is Big O Notation?

Big O notation is a mathematical notation used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity. In the context of algorithm analysis, it’s used to describe how the running time or space requirements of an algorithm scale with the size of the input.

The “O” in Big O notation stands for “order of magnitude,” which represents the rate of growth of the function as the input size increases. It’s important to note that Big O notation doesn’t provide an exact measurement of the running time or space usage of an algorithm, but rather an upper bound on these values.

## Why is important?

Big O notation is important because it allows us to compare the efficiency of different algorithms and choose the best one for a particular task. By analyzing the time and space complexity of algorithms, we can identify bottlenecks and optimize performance.

For example, let’s say we have two algorithms that perform the same task. One algorithm has a time complexity of O(n), while the other has a time complexity of O(n^2). If we’re working with small inputs, both algorithms might perform similarly, but as the input size grows, the O(n^2) algorithm will quickly become prohibitively slow. By using Big O notation, we can make an informed decision about which algorithm to use for a particular task.

## How is calculated?

To calculate the Big O notation of an algorithm, we first need to determine its basic operations. These are the fundamental steps that the algorithm performs, such as comparisons, assignments, and arithmetic operations.

Once we’ve identified the basic operations, we can count how many times each operation is performed as a function of the input size. For example, if an algorithm performs a single comparison for each element in an array, we can say that it has a time complexity of O(n), where n is the size of the array.

In general, we use the following rules to calculate the Big O notation of an algorithm:

- We ignore constant factors. For example, if an algorithm performs 2n comparisons, we can still say that it has a time complexity of O(n).
- We ignore lower-order terms. For example, if an algorithm performs n^2 + n comparisons, we can say that it has a time complexity of O(n^2).
- Worst case is usually the way we measure.

By applying these rules, we can simplify the time and space complexity of an algorithm to its essential features and express them using Big O notation.

## Common notation classes

There are several common Big O notation classes that are used to describe the time and space complexity of algorithms. Here are some of the most important ones:

- O(1): constant time complexity. The running time of the algorithm doesn’t depend on the size of the input.
- O(log n): logarithmic time complexity. The running time of the algorithm grows logarithmically with the size of the input.
- O(n): linear time complexity. The running time of the algorithm grows linearly with the size of the input.
- O(n log n): linearithmic time complexity. The running time of the algorithm grows as the product of the input size and the logarithm of the input size.
- O(n^2): quadratic time complexity. The running time of the algorithm grows quadratically with the size of the input.
- O(2^n): exponential time complexity. The running time of the algorithm grows exponentially with the size of the input.
- O(n!): factorial time complexity. The running time of the algorithm grows factorially with the size of the input.

## Conclusion

Big O notation is a powerful tool for analyzing the efficiency of algorithms and understanding how their performance changes as the input size grows. By using Big O notation, we can compare the relative efficiency of different algorithms and choose the most appropriate one for our needs. It’s important to keep in mind that Big O notation only describes the upper bound of an algorithm’s running time, and that the actual running time may be much faster in practice.

## Resources

The Last Algorithms Course You’ll Need is a comprehensive course on algorithm design and analysis, taught by ThePrimeagen, a software engineer with over 10 years of experience in the industry. The course covers a wide range of topics, from basic data structures to advanced algorithms, and provides practical examples and exercises to help you apply these concepts in real-world scenarios.

Throughout the course, you’ll learn how to analyze the time and space complexity of algorithms using Big O notation, and how to choose the best data structures and algorithms for a given problem. You’ll also learn about common algorithms and data structures such as arrays, linked lists, stacks, queues, trees, heaps, hash tables, and graph algorithms.