Big O Notation Explained Simply Using some Illustrations and a Video

If you‘re a programmer or computer science student, chances are you‘ve heard of Big O notation. It comes up in interviews, in university classes, and in hushed conversations between developers discussing their code. But what exactly is Big O notation, and why is it so important?

In simple terms, Big O notation is a way to measure how fast an algorithm is. More specifically, it describes the worst-case scenario, or the maximum time an algorithm will take to complete, in a way that is independent of the programming language or hardware used.

But don‘t let the name or the mathematical symbols intimidate you. While it might seem complex at first, you can learn to understand and apply Big O notation pretty quickly. It just takes a little practice to start thinking about your code in terms of Big O.

Common Big O Notations

There are several common Big O notations that every programmer should know:

O(1) – Constant Time
O(1) means that an operation will always take the same amount of time, regardless of the size of the input. This is the ideal scenario, as the running time of the algorithm does not increase as the input grows larger.

An example of an O(1) operation is accessing an array element by its index. No matter how large the array is, accessing a single element takes the same amount of time.

O(n) – Linear Time
An algorithm with O(n) time complexity has a running time that increases linearly with the size of the input. If you double the size of the input, the running time will roughly double.

Looping through the elements in an array is typically an O(n) operation. If the array has n elements, the loop will run n times.

O(log n) – Logarithmic Time
Algorithms with a time complexity of O(log n) have a running time that increases by a constant amount each time the size of the input is doubled. Logarithmic algorithms are super efficient for large datasets.

The most common example of an O(log n) algorithm is a binary search, where a sorted array is repeatedly divided in half to find a target value.

To understand why binary search is O(log n), consider how many times we can divide n by 2 until we get down to 1:

n	Number of divisions
1	0
2	1
4	2
8	3
16	4
32	5

In general, this is log₂n divisions. Therefore, the time complexity of binary search is O(log n).

O(n^2) – Quadratic Time
An algorithm with quadratic time complexity has a running time proportional to the square of the size of the input. These algorithms are significantly less efficient than a linear time algorithm and are often seen in nested loops.

A common example is a brute-force algorithm to find all pairs of elements in an array, which uses two nested loops.

There are a few other key Big O times to be aware of:

• O(n log n): Quasilinear time, worse than linear but better than quadratic. Sorting algorithms like Merge Sort and Quick Sort have a typical time complexity of O(n log n).
• O(2^n): Exponential time, which becomes very inefficient very quickly as the input grows. The recursive calculation of Fibonacci numbers is O(2^n) if implemented without memoization.
• O(n!): Factorial time, extremely inefficient for even small inputs. A brute-force solution to the traveling salesman problem takes O(n!) time.

Amortized Time Complexity

In some cases, an algorithm‘s time complexity may vary based on the specific operation being performed or the current state of the data structure. This is where the concept of amortized time complexity comes in.

Amortized time complexity refers to the average time taken per operation, over a sequence of operations. While a single operation might take a long time, if it occurs rarely and other operations are fast, the average time per operation can still be low.

A good example of this is a dynamic array (like a vector in C++ or an ArrayList in Java). Appending an element to the end of the array is usually an O(1) operation. But when the array reaches its capacity and needs to resize, the resize operation is O(n) as all elements need to be copied to a new, larger array.

However, because resizing happens infrequently (usually when the size doubles), the amortized time complexity for appending an element is still O(1). The cost of the occasional O(n) operation is "amortized" over all the cheap O(1) operations.

Big O and Common Data Structures

Understanding the time and space complexity of common data structures is crucial for writing efficient code. Here‘s a quick overview:

• Array: O(1) access, O(n) search, O(n) insertion/deletion
• Linked List: O(n) access, O(n) search, O(1) insertion/deletion at beginning or end
• Stack/Queue: O(1) insertion (push/enqueue) and deletion (pop/dequeue)
• Hash Table: O(1) average case for search, insertion, deletion; O(n) worst case
• Binary Search Tree: O(log n) average case for search, insertion, deletion; O(n) worst case
• Heap: O(log n) insertion and deletion; O(1) to access max/min element

Of course, these are just general guidelines. The actual complexities can vary based on the specific implementation and the balance of the data structure.

Big O in Practice

So how much does Big O actually matter in practice? Can understanding time complexity really make your code faster?

Absolutely. Consider this real-world example: Google Maps uses the Dijkstra algorithm with a heap data structure to calculate the shortest path between two points. The time complexity of this algorithm is O((E + V) log V), where E is the number of edges and V is the number of vertices in the graph.

By using this efficient algorithm and data structure, Google Maps is able to quickly calculate routes for millions of users every day. If they used a less efficient algorithm, like one with a time complexity of O(V^2), the service would grind to a halt under the load.

Here are some more examples of how Big O impacts real-world performance:

• Facebook uses a probabilistic data structure called a Bloom filter to efficiently check if a URL has been shared before. This data structure has a space complexity of O(n) and a time complexity of O(k) for insertions and lookups (where k is the number of hash functions), allowing Facebook to handle massive amounts of data.
• The Linux kernel uses a self-balancing binary search tree called a red-black tree for its process scheduler. This data structure provides O(log n) insertion, deletion, and search, ensuring efficient scheduling even with thousands of processes.

Tips for Full-Stack Developers

As a full-stack developer, you‘re likely to encounter algorithms and data structures at every level of the stack. Here are a few tips to keep in mind:

1. Think about scalability from the start. Even if your application is small now, design your algorithms and data structures with growth in mind. Use Big O to guide your choices.

2. Profile your code to find bottlenecks. Tools like Chrome DevTools or Python‘s cProfile module can help you identify which parts of your code are taking the most time. Focus your optimization efforts there.

3. Don‘t prematurely optimize. While it‘s important to consider efficiency, don‘t let it paralyze you. Write clear, maintainable code first, and then optimize the critical parts as needed.

4. Know your libraries and frameworks. Many common tasks, like sorting or searching, are already implemented efficiently in standard libraries. Use these when possible instead of reinventing the wheel.

5. Consider the total system complexity. Sometimes, a less efficient algorithm might be the right choice if it simplifies the overall system. For example, a O(n) algorithm that can be parallelized might be preferable to a O(log n) algorithm that can‘t.

Conclusion

Big O notation is a crucial tool for analyzing and comparing the efficiency of algorithms. It allows us to express how the running time or space requirements of an algorithm scale as the input size becomes arbitrarily large.

Understanding Big O is essential for several reasons:

• It helps us make informed decisions about which data structures and algorithms to use for a given problem.
• It allows us to predict how our code will perform as the amount of data grows, preventing surprises and ensuring scalability.
• It‘s a common language for communicating about code efficiency, which is especially important when working on a team or contributing to open source projects.
• It‘s frequently asked about in technical interviews, so a solid grasp of Big O can help you land your dream job.

But beyond these practical reasons, learning Big O is also about developing a mindset of efficiency and scalability. It trains you to think about the big picture, to consider the trade-offs between different approaches, and to always be asking "How can we do this better?"

So whether you‘re a student just starting out with algorithms, a full-stack developer looking to level up your skills, or a seasoned engineer tasked with optimizing a complex system, the concept of Big O is a powerful addition to your mental toolbox.

Keep practicing, keep analyzing, and most importantly, keep pushing yourself to write the most efficient, scalable code you can. Your future self (and your future users) will thank you.

For a more visual explanation of Big O notation, check out this excellent video tutorial:

Happy coding!