Decrease Key and Delete Node Operation on a Fibonacci Heap Data Structure  Data Structure Tutorial
Decrease Key and Delete Node Operation on a Fibonacci Heap Data Structure.
In this article, you will understand the Fibonacci operations of the Decrease key and delete the node.
It is a data structure that consists of a collection of trees that have a property of min heap or max heap property. These two properties are fundamental characteristics of the trees present on a Fibonacci heap. In the Fibonacci heap, the node can consist of more than two children and no children at all. It has more efficient heap operations than that supported by the binomial and binary heaps. It is known as a Fibonacci heap because the trees are constructed in a way such that a tree of order n has at least F(n+2) nodes in it.
Operations of Fibonacci Heap
 Decrease a key: It means we can decrease the value of a key to any lower value.
 Delete a node: It deletes the given node.
Decreasing a key
In decreasing a key operation, the value of a key is decreased to a lower value.
Functions used for decreasing the key:

DecreaseKey
 Choose the node whose value is to be decreased, y, and change its value to the new value x.
 If the value of the parent of y, i is not null and the key of the parent must be larger than that of the x then call Cut(y) and CascadingCut(i).
 If the key of y is lesser than the key of min, then mark y as min.

Cut
 Delete y from the present position and add it to the current root list.
 If y is marked, then mark it as false.

CascadingCut
 If the parent of i is not null then follow the next steps.
 If i is unmarked, then mark i.
 Otherwise, call Cut(i) and CascadingCut(parent of i).
Deleting a Node
It uses the decreasekey and extractmin operations. The steps involved in this are:
 Let y be the node to be deleted.
 Apply decreasekey operation to decrease the value of y to the lowest possible value.
 Apply extractmin operation to delete this node.
Complexity
The decreased key case complexity is O(1).
The Delete node case complexity is O(log n).