# What does $O(\alpha(n))$ amortized time mean?

DELETE(S, i): Delete integer $$i$$ from the set $$S$$. if $$i \notin S$$, there is no effect.

from a set of consectutive integers like $$S = \{1,2,3,5,6\}$$

Provide a data structure and an algorithm for DELETE that takes $$O(\alpha(n))$$ amortized time

not sure what what does $$O(\alpha(n))$$ amortized time mean?

I was thinking AVL trees ? I know the worst case is $$O(\log n)$$ for that. Not sure what $$O(\alpha(n))$$ amortized time means though.

$$\alpha$$ indicates the Inverse Ackermann Function: $$\alpha(n)$$ is the number such that $$A(\alpha(n), \alpha(n)) = n$$.
In practice, $$\alpha(n) \lt 5$$ for any input less than about $$^72$$. So $$O(\alpha(n))$$ is basically $$O(1)$$. The difference only matters in theory, never really in practice. You'll find $$O(\alpha(n))$$ mostly when working with disjoint sets, which are pretty much the fastest data structures you'll ever use.
EDIT: $$^nb$$ indicates "tetration", that is, iterated exponents. It grows ridiculously quickly: $$O(^n2)$$ is worse than $$O(n!)$$.
$$\alpha(n)$$ is the inverse Ackerman function.