What is Big O of a loop?

Question

I was reading about Big O notation. It stated,

The big O of a loop is the number of iterations of the loop into number of statement

Answer 1

First, don't call it "the Big O". That is wrong and misleading. What you are really trying to find is asymptotically how many instructions will be executed as a function of n. The right way to think about O(n) is not as a function, but rather as a set of functions. More specifically:

O(n) is the set of all functions f(x) such that there exists some constant M and some number x_0 where for all x > x_0, f(x) < M x.

In other words, as n gets very large, at some point the growth of the function (for example, number of instructions) will be bounded above by a linear function with some constant coefficient.

Depending on how you count instructions that loop can execute a different number of instructions, but no matter what it will only iterate at most n times. Therefore the number of instructions is in O(n). It doesn't matter if it repeats 6n or .5n or 100000000n times, or even if it only executes a constant number of instructions! It is still in the class of functions in O(n).

To expand a bit more, the class O(n*2) = O(0.1*n) = O(n), and the class O(n) is strictly contained in the class O(n^2). As a result, that loop is also in O(2*n) (because O(2*n) = O(n)), and contained in O(n^2) (but that upper bound is not tight).