确定算法的时间复杂度

Question

I'm reading one cool book, and there was a task to determine the time complexity of the code in the worst case. The code performs integer division (a > 0, b > 0):

int div(int a, int b) {
    int count = 0;
    int sum = b;
    while (sum <= a) {
        sum += b;
        count++;
    }
        
    return count;
}

My answer was O(n), since, in the worst case, when we have a > 0, b = 1, we will iterate through a times in the while() loop.

The answer in the book was O(a/b) which makes sense. But, if we're considering the worst case, then

O(a/1) => O(a) <=> O(n) (letter in parentheses doesn't really matter in this case).

So, the answer O(n) is correct as well. Isn't it?

Answer 1

Your answer does not make sense for the simple reason that there is no n in the problem statement, and saying that a is n is "cheating".

Also, though it is technically correct, it is a little sad to use an untight bound O(a) when you know that the algorithm makes exactly a/b additions.

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By the way, it is an arbitrary decision to say that b=1 is "the" worst case. Better say worst case for fixed a (and variable b), or for fixed b (and variable a).

Answer 2

Alas there is no generalisation of big-O to multi-variable one, see Formal definition of big-O when multiple variables are involved? .

But as you know the exact complexity, T(a,b)=a/b, there is no need for worst-case... We use worst-case analysis when we don't know how to compute the exact complexity.