I am trying to understand how the input size, problem size and the asymptotic behavior of an arbitrary algorithm given in pseudo code format differ from each other. While I fully understand the input and the asymptotic behavior, I have problems understanding the problem size. To me it looks as if problem size = space complexity for a given problem. But I am not sure. I'd like to illustrate my confusion with the following example:
We have the following pseudo code:
ALGONE(x,y) if x=0 or x=y then return 1 end return ALGONE(x-1,y-1) + ALGONE(x,y-1)
So let's say we give two inputs in $x$ and $y$ and $n$ represents the number of digits.
Since we are having addition as our main operation, and addition is an elementary operation, and for two numbers of $n$ digits, we need $n$ operations, then the asymptotic behavior is of the form $O(n)$.
But what about the problem size in this case. I don't understand what am I supposed to say. The term problem-size is so vague. It depends on the algorithm but even then, even if one is able to understand the algorithm what do you give as an answer?
I'd assume that in this particular case the problem size, might be the number of bits we need to represent the input. But this is a guess of mine, grounded in nothing.