Mike Vincent

Posted on Feb 17

How To Remember Big O Notation

#dsa #interview #bigonotation #webdev

Big O is everywhere, from coding to daily life. The key to remembering it? Know the patterns first, then practice spotting them.

Introduction to Big O Notation

Ever tried stacking chairs? Sure you have. One chair, easy. Two? Still easy. But what if every new chair had to be picked up, turned over twice, spun like a pizza, then finally stacked? Now you’re working.

Big O notation tells you how much work your code does as the job gets bigger. Some code runs smooth, no matter how much you throw at it. Other code? Starts strong, slows down, then grinds to a halt.

It’s everywhere. Finding a book in a library? That’s Big O. Sorting a deck of cards? Also Big O. Walking down the street, checking every house number till you find yours? You guessed it—Big O again.

Most people hear “Big O” and their brain locks up. Looks like math. Smells like math. But you don’t need to know everything. Just a few common patterns. Learn those, and you’ll start spotting them everywhere—at work, in code, in real life.

So let’s break it down. Let’s see what’s fast, what’s slow, and what turns your code into a train wreck when it gets too big.

Understanding Big O Notation: A Mathematical Tool for Algorithm Analysis

Big O notation is how you size up an algorithm. How fast it runs. How bad it slows down when the workload piles up. Some stay quick, no matter how much data you throw at them. Others? They start gasping for air. Big O tells you which is which.

The Basics of Big O Notation

Big O notation looks like this: O(f(n)). That f(n)? It’s just a way to say, "What happens when n gets big?" In plain English, Big O tells you the worst-case scenario—how much time or space an algorithm will need when the input keeps growing. The bigger the job, the more it matters.

Key Components of Big O Notation:

O: Represents the upper bound or worst-case scenario.
f(n): A function that describes how the algorithm's performance grows with respect to the input size n.

Why Do We Use Big O Notation?

When you break down an algorithm, you care about two things—time and space. How long it takes. How much memory it eats. Big O notation gives you a way to put numbers on both, so you know what you’re dealing with before it’s too late.

Two Important Aspects of Algorithm Analysis:

Time Complexity: The number of operations an algorithm needs to process input data.
Space Complexity: The amount of memory required to run the algorithm.

The Formal Definition of Big O Notation

The formal definition goes like this: a function f(n) belongs to O(g(n)) if there are positive constants c and n₀ such that:

0 ≤ f(n) ≤ c × g(n) for all n ≥ n₀

In plain terms? Once n gets big enough, f(n) won’t grow faster than some multiple of g(n). Big O puts a cap on how wild things can get.

Simplifying Functions Using Big O Notation

Big O only cares about the biggest thing that changes as the problem gets bigger. It ignores small numbers and extra steps because they don’t matter much when things grow huge. It’s like knowing that in a race, the fastest runner decides who wins—not who ties their shoes the quickest.

Examples of Function Simplification:

2n² + 3n + 1 simplifies to O(n²)
n + 100 simplifies to O(n)
5 simplifies to O(1)

Predicting Algorithm Behavior with Asymptotic Analysis

Big O helps us see what happens when data gets big—really big. Some algorithms start fast but slow down as the workload grows. Others stay quick no matter what.

Say you have an algorithm with O(n²). Feels fine with small numbers. But double the data, and it takes four times as long. Keep going, and soon it's too slow to use. That’s why we check Big O—so we know what will last and what will crash.

Comparing Algorithms Using Big O Notation

Big O gives us a way to compare algorithms. Which one is faster? Which one takes less space? Instead of guessing, we use Big O to see the numbers.

When you have different ways to solve a problem, Big O helps you pick the best one. A slow algorithm might work for small tasks but fall apart when things get big. A faster one might take more memory but run in a snap.

Knowing Big O helps programmers write better code, speed things up, and make smarter choices when building software.

Common Time Complexities Made Easy

Algorithms follow patterns. Some stay fast, some slow down, and some collapse under too much data. Knowing these patterns helps us understand how code holds up as things grow.

Let’s start with the simplest one—constant time. No matter how much data you have, it runs in the same amount of time. Quick, predictable, and the best-case scenario.

1. Constant Time Complexity O(1)

Think of a vending machine. You press a button, and out comes your snack. Doesn’t matter if the machine is full or almost empty—the time it takes stays the same. That’s constant time complexity (O(1)).

Same idea in code:

def get_first_item(array):
    return array[0]  # Always takes the same time

Whether the array has 10 items or 10 million, grabbing the first one is instant. The algorithm doesn’t scan the whole list—it just reaches in and pulls out the answer.

Other examples of O(1):

Looking up a value in a hash table
Adding two numbers
Checking if a number is odd or even
Pushing to a stack
Reading an array value at a fixed index

The magic of O(1)? It never slows down. The work stays the same no matter how big the data gets.

Think of it like a fixed-price menu at a restaurant. Order one dish or ten—the price stays the same. That’s why constant time is the best-case scenario in programming.

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

Logarithmic Time: The Art of Cutting Work in Half

Ever looked up a name in an old-school phone book? You don’t flip through every page. You open it in the middle, check where you are, and cut the search space in half. Keep doing that, and you find the name in just a few steps—that’s logarithmic time (O(log n)) in action.

This trick shows up in fast algorithms:

Binary Search – Finding a number in a sorted list
Binary Trees – Searching through well-structured data
Divide and Conquer – Breaking big problems into smaller ones

The best part? It scales beautifully. When the input doubles, you only need one extra step to handle it.

8 items → 3 steps
16 items → 4 steps
32 items → 5 steps

How to spot O(log n):

The problem keeps getting chopped in half
You’re working with sorted or tree-based structures
The workload shrinks fast

Logarithmic time is what makes efficient searching possible. Instead of drowning in data, you slice through it.

3. Linear Time Complexity (O(n))

Linear time complexity (O(n)) means the work grows directly with the input. Double the data? Double the time.

Picture a teacher taking attendance. One student, one check. If the class has 10 students, the teacher calls 10 names. 100 students? 100 checks. The bigger the group, the longer it takes.

Same thing happens in code. Say you’re searching for a name in an unsorted list. You start at the top and check each one until you find it.

10 names? Up to 10 checks
100 names? Up to 100 checks

This one-by-one search is why linear time is easy to spot in code:

O(n) is practical for small to medium inputs, but when data piles up, it slows down. It’s faster than quadratic time but can’t compete with constant or logarithmic time. If you see a loop that runs through every element, chances are you’re looking at linear time.

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

O(n log n) is what happens when you split and conquer. Instead of tackling a big problem all at once, you break it into pieces, solve each piece, then put it back together. It’s the secret behind fast sorting.

Think about sorting a deck of cards using merge sort. You split the deck in half again and again—this takes O(log n). Then you merge the sorted pieces back together—this takes O(n). Multiply the two, and you get O(n log n).

This is why many efficient sorting algorithms like merge sort, heap sort, and quicksort use O(n log n). They’re way faster than O(n²) sorting methods because they use smart divide-and-conquer strategies.

It’s like organizing books in a library:

Split them into smaller groups (O(log n))
Sort each group (O(n))
Merge everything back together (O(n))

Instead of comparing every book to every other book, you divide and sort efficiently.

You’ll see O(n log n) in:

Sorting large datasets
Building optimal binary search trees
Finding closest pairs of points
Performing Fast Fourier Transforms

When you need speed and structure, O(n log n) is your best bet.

5. Quadratic Time Complexity (O(n²))

Quadratic time complexity happens when an algorithm’s running time grows with the square of the input size. The bigger the input, the worse it gets.

Picture arranging 100 books on a shelf. You don’t just place them—you compare each book with every other book to find its correct position. That means 100 × 100 = 10,000 comparisons.

A classic example is bubble sort. For a list of 5 numbers:

The first pass compares 4 pairs
The second pass compares 3 pairs
The total comparisons add up to n × (n - 1) / 2, which simplifies to O(n²).

Identifying Quadratic Time Complexity

A clear warning sign? Nested loops.

for i in range(n):  
    for j in range(n):  
        # perform operation

You’ll find O(n²) in algorithms that:

Compare each element with every other element
Build multiplication tables
Check all possible pairs in a dataset
Find duplicate elements in an unsorted array

Impact of Quadratic Time Complexity

Quadratic algorithms don’t scale well. If you have 1,000 elements, that means 1,000,000 operations. That’s a massive slowdown compared to linear or logarithmic alternatives.

This is why understanding algorithm complexity is key—so you know when to avoid O(n²) and pick something faster.

Beyond Quadratic: Higher Complexities Like Exponential (O(2^n)) and Factorial (O(n!))

Exponential time complexity O(2^n) creates a dramatic increase in processing time as input size grows. A classic example is the brute-force solution to the traveling salesman problem - visiting n cities requires checking 2^n possible routes.

Factorial time complexity O(n!) presents an even steeper growth curve. Consider generating all possible arrangements of n items. With just 10 items, you need to process 3,628,800 combinations. At 20 items, the number skyrockets to roughly 2.4 quintillion combinations.

These algorithms become impractical for real-world applications. A program with O(2^n) complexity processing 50 items would take years to complete on modern hardware. For O(n!) complexity, even 15 items could overwhelm most systems.

Practical Impact:

O(2^n): Doubles processing time for each additional input
O(n!): Multiplies processing time by each increasing number
Both complexities limit practical applications to very small input sizes

Real-World Connections: How Algorithms Shape Everyday Life

Algorithms aren’t just for computers. They show up in everyday life, from finding a book to sorting mail. Different tasks follow different time complexities—some are fast, some slow down as the workload grows.

Searching for a Book in a Library

If you know exactly where the book is, you walk straight to the shelf and grab it. That’s O(1), constant time.

If you don’t know the location, you might use the library’s computer system to search by title. The system sorts books like a dictionary, cutting the search space in half with each step. That’s O(log n), logarithmic time.

Walking Down a Street

Imagine checking house numbers one by one until you find your destination. The longer the street, the longer it takes. That’s O(n), linear time.

Organizing a Closet

Sorting clothes? You might first divide them by type, then sort by color. That’s O(n log n), linearithmic time.

But if you compare every piece of clothing with every other piece to sort them? That’s O(n²), quadratic time—much slower.

Grocery Store Checkout

Real-world tasks often mix time complexities. A grocery store checkout has:

Constant time (O(1)) for scanning a single item
Linear time (O(n)) for checking out all the items in your cart
Logarithmic time (O(log n)) if the system quickly looks up a price in a sorted database

Postal Service

The mail system uses different efficiencies at different steps:

Searching a sorted batch of mail? That’s binary search, O(log n).
Delivering mail door-to-door? That’s O(n), since each house gets a letter.
Sorting unsorted mail into routes? That’s O(n log n), using efficient sorting methods like merge sort.

These examples show that algorithms aren’t just abstract math. They’re baked into how we organize, search, and process tasks every day.

Worst-case vs. Average-case: What Really Matters?

When analyzing an algorithm, you don’t just look at how fast it can be—you also consider how slow it might be. Worst-case and average-case complexities help programmers choose the right approach for the job.

Worst-case Complexity

This is the longest time an algorithm could take to finish. Imagine searching for a number in an unsorted list. Worst case? The number is at the very end, meaning you have to check every element. That’s O(n).

Average-case Complexity

This is how the algorithm typically performs across different inputs. In the same unsorted list, you might find the number somewhere in the middle, meaning fewer checks on average.

Take QuickSort as an example:

Worst-case: O(n²) if the pivot selection is bad. More details here.
Average-case: O(n log n), which is why QuickSort is widely used. More on that here.

Why It Matters

Some applications must guarantee fast performance, even in the worst case. Others are fine if they usually run fast but occasionally slow down. That’s why sorting algorithms make trade-offs—some have very different worst- and average-case complexities.

Knowing both helps programmers choose wisely—balancing reliability with speed where it matters most.

Time and Space Tradeoffs in Algorithms: Finding the Right Balance for Optimal Performance

Optimizing an algorithm is often a balancing act between speed and memory. A faster algorithm might need more memory, while a memory-efficient solution could slow things down.

This tradeoff shapes how algorithms are designed for real-world use. Some applications need speed at any cost, while others must work within tight memory limits. The challenge is finding the right balance based on the needs of the task.

Understanding the Tradeoff: An Example

Imagine a phone app that stores user contacts. If speed is the top priority, the app can keep all contacts in memory. That way, searching for a name happens instantly. But there’s a cost—this method eats up system memory, which can be a problem on devices with limited resources.

Now, let’s flip the priority. Instead of storing everything in memory, the app only loads contacts when needed. This saves memory, but there’s a catch—reading from storage is slower than fetching from memory, so searches take longer.

This is the classic tradeoff. One approach is fast but uses more memory. The other saves memory but sacrifices speed. The right choice depends on what matters more for the app.

Use Cases: When to Optimize for Time or Space

The decision between optimizing for time or space depends on the specific use cases:

Memory-Critical Systems: In scenarios where devices have limited RAM, such as embedded systems or IoT sensors, space efficiency becomes paramount.
Time-Sensitive Applications: Applications that require immediate responses or real-time processing, like trading systems or online gaming platforms, prioritize speed over memory usage.
Balanced Solutions: Most web applications fall somewhere in between these two extremes and require a compromise between time and space optimization.

Sorting Algorithms: A Practical Example

Sorting algorithms are a great example of how performance and resource use can clash. QuickSort is efficient and doesn’t need much extra memory, making it a popular choice. But in some cases, it slows down compared to other methods.

MergeSort, on the other hand, runs consistently fast no matter what. The catch? It needs extra memory to store temporary arrays while sorting.

This is the kind of tradeoff programmers face all the time—choosing between speed and resource use, depending on what the task demands.

The Impact of Cloud Computing on Tradeoffs

Cloud computing has shifted the way developers think about speed and memory. Cloud platforms offer scalable resources, meaning applications can get more memory or processing power as needed. This allows developers to focus on speed without stressing over memory limits because the cloud provider handles it.

But this isn’t true everywhere. In mobile development, resources are fixed. Devices have limited memory, and battery life is a concern. A memory-hungry app can drain power or crash a device. Developers must be careful about how much data they load at once to keep performance smooth.

The lesson? Cloud apps can afford to optimize for speed, but mobile apps still need to be lean.

The Importance of Understanding Time and Space Complexity

These tradeoffs show why developers need to understand both time and space complexity when designing algorithms. Big O notation helps compare different approaches based on how much time they take and how much memory they use.

By looking at both factors together, developers can choose the right algorithm for the job. Some applications need to be as fast as possible, while others must fit within strict memory limits. Knowing how to balance these constraints leads to better, more efficient solutions.

Tips for Mastering Big O Notation: From Memorization Techniques to Practical Application Strategies

1. Use the Staircase Method for Memorization

A simple way to remember Big O notation is through the "Staircase Method." Picture a staircase where each step represents a different time complexity, arranged from fastest to slowest:

O(1) → Ground floor (constant) O(log n) → First step (logarithmic) O(n) → Second step (linear) O(n log n) → Third step (linearithmic) O(n²) → Fourth step (quadratic) O(2^n) → Fifth step (exponential) O(n!) → Top floor (factorial)

2. Apply the Loop Counter Technique

Another helpful technique is the "Loop Counter":

No loops = O(1)
Single loop = O(n)
Nested loop = O(n²)
Divide input = O(log n)

3. Recognize Patterns Visually

Visual Pattern Recognition:

Think of O(1) as a straight line
O(log n) looks like a curved line that flattens
O(n) appears as a diagonal line
O(n²) resembles a steep curve upward

4. Practice Identifying Patterns

Practice Pattern:

Identify the loops in your code
Check if the input is being divided
Look for nested operations
Count the operations that grow with input size

These patterns become second nature with practice. Try analyzing simple code snippets daily to build pattern recognition skills.

Wrapping Up: Big O Notation

Big O is a must-know for programmers. It helps you pick the right algorithm, speed up your code, and handle big data without breaking things.

And this isn’t just theory. In real life, fast code matters—search engines handle millions of searches, apps run on limited power, and slow programs waste time and money. Knowing O(1), O(log n), O(n), and O(n²) lets you predict how your code will hold up when the workload grows.

Start small. Look at the code you write every day. Can you make it faster? Can you use a better data structure? The goal isn’t always to chase the lowest Big O but to find a smart balance between speed, simplicity, and keeping your code easy to read.

The more you understand Big O, the better your software scales. And when your programs stay fast, no matter how much data they handle, you’re doing it right.

FAQs (Frequently Asked Questions)

What is Big O notation?

Big O notation is a mathematical concept used to describe the performance characteristics of algorithms, specifically in terms of time and space complexity. It helps in analyzing how the execution time or memory requirements of an algorithm grow as the input size increases.

Why is understanding Big O notation important?

Understanding Big O notation is crucial for programmers and computer scientists as it allows them to evaluate the efficiency of algorithms. It aids in making informed decisions about which algorithms to use based on their performance in different scenarios, especially as data sizes grow.

What are some common time complexities in Big O notation?

Some common time complexities include: Constant Time (O(1)), Logarithmic Time (O(log n)), Linear Time (O(n)), Linearithmic Time (O(n log n)), Quadratic Time (O(n²)), Exponential Time (O(2^n)), and Factorial Time (O(n!)). Each represents a different growth rate in execution time relative to input size.

How can I remember different types of time complexities?

An easy way to remember different types of time complexities is by associating them with relatable examples and visual aids. Mnemonic devices can also help reinforce your understanding. For instance, think of O(1) as a light switch that turns on instantly, while O(n²) can be visualized as nested loops where each iteration depends on the previous one.

What is the difference between worst-case and average-case complexities?

Worst-case complexity refers to the maximum amount of time or resources an algorithm will require for any input size, while average-case complexity provides an estimate of performance considering all possible inputs. Analyzing both gives a balanced perspective on algorithm performance.

How can algorithms be optimized for performance?

Algorithms can be optimized for performance by balancing execution speed and memory usage. Techniques include choosing more efficient algorithms, reducing unnecessary computations, and utilizing data structures that minimize resource consumption. Understanding the trade-offs between time and space complexity is key to achieving optimal performance.

Next Steps for Big O

Big O isn’t just for code—it’s everywhere. The more you spot it, the better you’ll remember it.

Find Big O in Daily Tasks

Sorting laundry? That’s O(n log n). Checking every key to find the right one? O(n). Think about time complexity while you go about your day.

Compare Common Searches

Looking up a word in a dictionary is O(log n), but scanning a menu for your favorite dish is O(n). Spot the difference in real life.

Test Yourself Everywhere

Waiting in line? Sorting cards? Walking through a crowded mall? Ask yourself—how does this scale if the problem gets bigger?

The more you practice, the easier Big O will stick. Try it today.

How Do You Remember Big O?

Big O is key to writing fast code, but how do you keep it in mind? Do you quiz yourself, connect it to real life, or just learn by doing?

Think back—when was the last time your program slowed down? Did you stop to figure out why? Maybe you optimized a loop, maybe you didn’t notice until it was too late.

What’s your best trick for remembering Big O? Share your advice in the comments!

Mike Vincent is an American software engineer and writer based in Los Angeles. Mike writes about technology leadership and holds degrees in Linguistics and Industrial Automation. More about Mike Vincent