Question

Can someone please explain to me how one can determine the worst-case complexity of an algorithm. I know that the we need to use the equation W(n) = max{t(I)|I element of D), where D is the set of inputs of size n. Do I calculate the number of operations performed for each element I and then take its max? What easier way is there to accomplish this?

Answer 1

Starting from the equation is thinking of it a bit backwards. What you really care about is scalability, or, what is it going to do as you increase the size of the input.

If you just have a loop, for instance, you have a O(n) time complexity algorithm. If you have a loop within another loop though, it becomes O(n^2), because it must now do n^2 many things for any size n input.

When you are talking about worst case, you are usually talking about non deterministic algorithms, where you might have a loop that can stop prematurely. What you want to do for this is assume the worst and pretend the loop will stop as late as possible. So if we have:

for(int i = 0;i < n;i++){
for(int j = 0;j < n;j++){
if(rand() > .5) j = n;
}
}

We would say that the worst-case is O(n^2). Even though we know that it is very likely that the middle loop will bust out early, we are looking for the worst possible performance.