Question

Some standard books on Algorithms produce this:

0 ≤ f(n) ≤ c⋅g(n) for all n > n₀

While defining big-O, can anyone explain to me what this means, using a strong example which can help me to visualize and understand big-O more precisely?

Answer 1

Assume you have a function f(n) and you are trying to classify it – is it a big O of some other function g(n).

The definition basically says that f(n) is in O(g(n)) if there exists two constants C,N such that

f(n) <= c * g(n) for each n > N

Now, let’s understand what it means.

Start with the n>N part – it means, we do not “care” for low values of n, we only care for high values, and if some (final number of) low values do not follow the criteria – we can silently ignore them by choosing N bigger then them.

Have a look on the following example:

Though we can see that for low values of n: n^2 < 10nlog(n), the second quickly catches up and after N=10 we get that for all n>10 the claim 10nlog(n) < n^2 is correct, and thus 10nlog(n) is in O(n^2).
The constant c means we can also tolerate some multiple by constant factor, and we can still accept it as desired behavior (useful for example to show that 5*n is O(n), because without it we could never find N such that for each n > N: 5n < n, but with the constant c, we can use c=6 and show 5n < 6n and get that 5n is in O(n).