I tried to make this clear in my comment - (asymptotic) "upper bound" is a concept applicable to any function from N to R. So do you mean the upper bound of the worst case, the upper bound of the average case, the upper bound of the best case, the upper bound of the worst case memory usage, the upper bound of the median price of eggs in China on the nth day since 1970, or something else?
Ya, I just mean that without further qualification, it's the upper bound of the whole function that's being referred to, which is equivalent to the upper bound of the worst case.
No, you still don't understand. The worst-case runtime and the average-case runtime are different functions. There's no one "whole function" that both are part of.
They are not different functions. They are the same function. The average case is a special case of the general function in which some of the terms collapse.
I don't know what to say at this point. Any reputable reference will disagree with you, including the two Stack Overflow links I posted, or even the Wikipedia article on big-O notation, which doesn't say anywhere that it has to do with the worst case of algorithms.
> They are not different functions. They are the same function.
Sure, if you expand the term "function" to include "random variable", which is legitimate to do in some fields of math... but we're talking about deterministic functions from the natural numbers (or real numbers) to the real numbers here. Which is what big-O notation is defined for.
Can you write down the deterministic function from N->R giving the "runtime" of some algorithm? (I mean the actual function, not big-O).
I tried to make this clear in my comment - (asymptotic) "upper bound" is a concept applicable to any function from N to R. So do you mean the upper bound of the worst case, the upper bound of the average case, the upper bound of the best case, the upper bound of the worst case memory usage, the upper bound of the median price of eggs in China on the nth day since 1970, or something else?