Introduction to Asymptotic Analysis and Big O

We have seen that the time complexity of an algorithm can be expressed as a polynomial. To compare two algorithms, we can compare the respective polynomials. However, the analysis performed in the previous lessons is a bit cumbersome and would become intractable for bigger algorithms that we tend to encounter in practice.

Asymptotic analysis

One observation that helps us is that we want to worry about large input sizes only. If the input size is really small, how bad can a poorly-designed algorithm get, right? Mathematicians have a tool for this sort of analysis called the asymptotic notation. The asymptotic notation compares two functions, say, $f(n)$ and $g(n)$ for very large values of $n$. This fits in nicely with our need for comparing algorithms for very large input sizes.

Big O notation

One of the asymptotic notations is the Big O notation. A function $f(n)$ is considered $O(g(n))$, read as big oh of $g(n)$, if there exists some positive real constant $c$ and an integer $n_0 \geq 0$, such that the following inequality holds for all $n \geq n_0$:

$f(n) \leq cg(n)$

The following graph shows a plot of a function $f(n)$ and $cg(n)$ that demonstrates this inequality.

Note that the above inequality does not have to hold for all $n$. For $n < n_0$, $f(n)$ is allowed to exceed $cg(n)$, but for all values of $n$ beyond some value $n_0$, $f(n)$ is not allowed to exceed $cg(n)$.

What good is this? It tells us that for very large values of $n$, $f(n)$ ...