## 1. Overview

In this brief tutorial, we’ll learn about **how big-O and little-o notations differ**. In short, they are both asymptotic notations that specify upper-bounds for functions and running times of algorithms.

However, the difference is that **big-O may be asymptotically tight** while **little-o makes sure that the upper bound isn’t asymptotically tight**.

Let’s read on to understand what exactly it means to be asymptotically tight.

## 2. Mathematical Definition

Big-O and little-o notations have very similar definitions, and **their difference lies in how strict they are regarding the upper bound** they represent.

### 2.1. Big-O

For a given function , is defined as:

there exist positive constants and such that for all .

**So** **is** **a set of functions** **that are, after** , **smaller than or equal to **. The function’s behavior before is unimportant since big-O notation (also little-o notation) analyzes the function for huge numbers. As an example, let’s have a look at the following figure:

Here, is only one of the possible functions that belong to . Before , is not always smaller than or equal to , but after , it never goes above .

**The equal sign in the definition represents the concept of** **asymptotical tightness**, **meaning that when** **gets very large,** **and** **grow at the same rate**. For instance, satisfies the equal sign, hence it is asymptotically tight, while is not.

For more explanation on this notation, look at an introduction to the theory of big-O notation.

### 2.2. Little-o

**Little-o notation is used to denote an upper-bound that is not asymptotically tight**. It is formally defined as:

for any positive constant , there exists positive constant such that for all .

**Note that in this definition**, **the set of functions **** are strictly smaller than** , **meaning that** **little-o notation is a stronger upper bound than big-O notation**. In other words, the little-o notation does not allow the function to have the same growth rate as .

Intuitively, this means that as the approaches infinity, becomes insignificant compared to . In mathematical terms:

In addition, the inequality in the definition of little-o should hold for any constant , whereas for big-O, it is enough to find some that satisfies the inequality.

If we drew an analogy between asymptotic comparison of and and the comparison of real numbers and , we would have while .

## 3. Examples

Let’s have a look at some examples to make things clearer.

For , we have:

- but
- and
- and

In general, for , we will have:

- but
- and
- and

For more examples of big-O notation, see practical Java examples of the big-O notation.

## 4. Conclusion

In this article, we learned the difference between big-O and little-o notations and noted that little-o notation excludes the asymptotically tight functions from the set of big-O functions.