# Input that causes an operation on a binomial heap to run in $\Omega(\log n)$ time?

I was studying binomial heaps and its time analysis. Are there any inputs that cause DELETE-MIN, DECREASE-KEY, and DELETE to run in $\Omega(\log n)$ time for a binomial heap rather than $O(\log n)$?

• On a small technical note, $T(n)\in \Omega(f(n))$ doesn't preclude $T(n) \in O(f(n))$. – Luke Mathieson Jan 16 '13 at 8:09

This paper shows that for each priority queue implementation satisfying some technical condition (which holds for binomial heaps) there is a sequence of operations consisting of $n$ INSERTs, $n$ DELETEs and $n$ FIND-MINs that takes time $\Omega(n\log n)$. Since INSERTs are amortized $O(1)$ in binomial heaps, the DELETEs and FIND-MINs together must take time $\Omega(n\log n)$. Unfortunately, in the construction the INSERTs are interspersed between the other operations.
This is of course overkill for the case of binomial heaps.In the implementation in which FIND-MIN is implemented by comparing the roots of all trees. If we INSERT $2^m-1$ elements then we will have $m = \log(n+1)$ trees, and so FIND-MIN will take time $\Omega(\log n)$. In other implementations, it shouldn't be hard to concoct examples in which some MERGE operation (used inside all the others) takes time $\Omega(\log n)$.
You might be worried that even though these examples show that some operations could require logarithmic time, the amortized complexity is still better. You'll have to work harder, but after some though you should be able to come up with a sequence of $n$ INSERTs and DELETEs which takes amortized time $\Omega(\log n)$.