Determining the complexities given codes

Question

Given a snipplet of code, how will you determine the complexities in general. I find myself getting very confused with Big O questions. For example, a very simple question:

Answer 1

I like to use two things for Big-O notation: standard Big-O, which is worst case scenario, and average Big-O, which is what normally ends up happening. It also helps me to remember that Big-O notation is trying to approximate run-time as a function of N, the number of inputs.

The TA explained this with something like combinations. Like this is n choose 2 = (n(n-1))/2 = n^2 + 0.5, then remove the constant so it becomes n^2. I can put int test values and try but how does this combination thing come in?

As I said, normal big-O is worst case scenario. You can try to count the number of times that each line gets executed, but it is simpler to just look at the first example and say that there are two loops over the length of n, one embedded in the other, so it is n * n. If they were one after another, it'd be n + n, equaling 2n. Since its an approximation, you just say n or linear.

What if theres an if statement? How is the complexity determined?

This is where for me having average case and best case helps a lot for organizing my thoughts. In worst case, you ignore the if and say n^2. In average case, for your example, you have a loop over n, with another loop over part of n that happens half of the time. This gives you n * n/x/2 (the x is whatever fraction of n gets looped over in your embedded loops. This gives you n^2/(2x), so you'd get n^2 just the same. This is because its an approximation.

I know this isn't a complete answer to your question, but hopefully it sheds some kind of light on approximating complexities in code.

As has been said in the answers above mine, it is clearly not possible to determine this for all snippets of code; I just wanted to add the idea of using average case Big-O to the discussion.