Min Fibonacci Heap - increase key

Question

I have been trying to implementing heap data structures for use in my research work. As part of that, I am trying to implement increase-key operations for min-heaps. I know that min-heaps generally support decrease-key. I was able to write the increase-key operation for a binary min-heap, wherein, I exchange increased key with the least child recursively. In the case of the Fibonacci heap, In this reference, they say that the Fibonacci heap also supports an increase-key operation. But, I couldn't find anything about it in the original paper on Fibonacci Heaps, nor could I find anything in CLRS (Introduction to Algorithms by Cormen).

Can someone tell me how I can go about implementing the increase-key operation efficiently and also without disturbing the data structure's amortized bounds for all the other operations?

Answer 1

First note that $\text{increase-key}$ must be $O(\log n)$ if we wish for $\text{insert}$ and $\text{find-min}$ to stay $O(1)$ as they are in a Fibonacci heap.

If it weren't you'd be able to sort in $O(n)$ time by doing $n$ $\text{insert}$s, followed by repeatedly using $\text{find-min}$ to get the minimum and then $\text{increase-key}$ on the head by $\omega$ with $\forall x:\omega > x$ to push the head to the end.

Now, knowing that $\text{increase-key}$ must be $O(\log n)$ we can provide a very simple asymptotically optimal implementation for it. To increase a node $n$ to value $x$, first you $\text{decrease-key}(n, -\infty)$, then $\text{delete-min()}$ followed by $\text{insert}(n, x)$.