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 g(n), O(g(n)) is defined as:

O(g(n)) = \{f(n): there exist positive constants c and n_0 such that 0 \le f(n) \le cg(n) for all n \ge n_0\}.

So \pmb{O(g(n))} is a set of functions that are, after \pmb{n_0}, smaller than or equal to \pmb{g(n)}. The function’s behavior before n_0 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, f(n) is only one of the possible functions that belong to O(g(n)). Before n_0, f(n) is not always smaller than or equal to g(n), but after n_0, it never goes above g(n).

The equal sign in the definition represents the concept of asymptotical tightness, meaning that when \pmb{n} gets very large, \pmb{f(n)} and \pmb{g(n)} grow at the same rate. For instance, 3n^3 = O(n^3) satisfies the equal sign, hence it is asymptotically tight, while 3n = O(n^3) 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:

o(g(n)) = \{f(n): for any positive constant c, there exists positive constant n_0 such that 0 \le f(n) < cg(n) for all n \ge n_0\}.

Note that in this definition, the set of functions \pmb{f(n)} are strictly smaller than \pmb{cg(n)}, 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 f(n) to have the same growth rate as g(n).

Intuitively, this means that as the n approaches infinity, f(n) becomes insignificant compared to g(n). In mathematical terms:

    \[ \lim_{n\to\infty} \frac{f(n)}{g(n)} = 0\]

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

If we drew an analogy between asymptotic comparison of f(n) and g(n) and the comparison of real numbers a and b, we would have f(n) = O(g(n)) \approx a \le b while f(n) = o(g(n)) \approx a < b.

3. Examples

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

For f(n) = 2n + 3, we have:

  • \pmb{f(n) = O(n)} but \pmb{f(n) \ne o(n)}
  • f(n) = O(n^2) and f(n) = o(n^2)
  • f(n) = O(n^3) and f(n) = o(n^3)

In general, for f(n) = a_xn^x + a_{x-1}n^{x-1} + ... + a_0, we will have:

  • \pmb{f(n) = O(n^x)} but \pmb{f(n) \ne o(n^x)}
  • f(n) = O(n^{x+1}) and f(n) = o(n^{x+1})
  • f(n) = O(n^{x+2}) and f(n) = o(n^{x+2})

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.

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