A visual introduction to big O notation

O(1) in many cases involves a hashing function which is a non-trivial but constant cost. For smaller values of N it can be outperformed in terms of wall clock time by n^2 worst case algorithms.

I mean, true obviously, but don't say that too loud lest people get the wrong ideas. For most practical purposes n^2 means computer stops working here. Getting people to understand that is hard enough already ;)

Besides, often you're lucky and there's a trivial perfect hash like modulo.

What do you mean? Modulo is not a perfect hash function... What if your hash table had size 11 and you hash two keys of 22 and 33?

I also don't understand your first point. We can run n^2 algorithms on massive inputs given its just a polynomial. Are you thinking of 2^n perhaps?

n^2 algorithms on _massive_ inputs seems a little far fetched, no?

Around one to one hundred billion things start getting difficult.