In the last post, we discussed the Insertion and Union of Fibonacci Heaps. In this post, we will discuss Extract_min(), Decrease_key() and Deletion() operations on Fibonacci heap.
Extract_min(): We create a function for deleting the minimum node and setting the min pointer to the minimum value in the remaining heap. The following algorithm is followed:
- Delete the min node.
- Set head to the next min node and add all the trees of the deleted node in the root list.
- Create an array of degree pointers of the size of the deleted node.
- Set degree pointer to the current node.
- Move to the next node.
- If degrees are different then set degree pointer to next node.
- If degrees are the same then join the Fibonacci trees by union operation.
- Repeat steps 4 and 5 until the heap is completed.
Decrease_key(): To decrease the value of any element in the heap, we follow the following algorithm:
- Decrease the value of the node ‘x’ to the new chosen value.
- CASE 1) If min-heap property is not violated,
- Update min pointer if necessary.
- CASE 2) If min-heap property is violated and parent of ‘x’ is unmarked,
- Cut off the link between ‘x’ and its parent.
- Mark the parent of ‘x’.
- Add tree rooted at ‘x’ to the root list and update min pointer if necessary.
- CASE 3)If min-heap property is violated and parent of ‘x’ is marked,
- Cut off the link between ‘x’ and its parent p[x].
- Add ‘x’ to the root list, updating min pointer if necessary.
- Cut off link between p[x] and p[p[x]].
- Add p[x] to the root list, updating min pointer if necessary.
- If p[p[x]] is unmarked, mark it.
- Else, cut off p[p[x]] and repeat steps 4.2 to 4.5, taking p[p[x]] as ‘x’.
Deletion(): To delete any element in a Fibonacci heap, the following algorithm is followed:
- Decrease the value of the node to be deleted ‘x’ to a minimum by Decrease_key() function.
- By using min-heap property, heapify the heap containing ‘x’, bringing ‘x’ to the root list.
- Apply Extract_min() algorithm to the Fibonacci heap.
Following is a program to demonstrate Extract min(), Deletion() and Decrease key() operations on a Fibonacci Heap:
Creating an initial heap The root nodes of Heap are: 2-->5-->8 The heap has 3 nodes Extracting min The root nodes of Heap are: 5 The heap has 2 nodes Decrease value of 8 to 7 The root nodes of Heap are: 5 The heap has 2 nodes Delete the node 7 Key Deleted The root nodes of Heap are: 5 The heap has 1 nodes
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