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# Decrease Key and Delete Node Operations on a Fibonacci Heap

#### In this tutorial, you will learn how decrease key and delete node operations work. Also, you will find working examples of these operations on a fibonacci heap in Python.

In a fibonacci heap, decrease-key and delete-node are important operations. These operations are discussed below.

## Decreasing a Key

In decreasing a key operation, the value of a key is decreased to a lower value.

Following functions are used for decreasing the key.

### Decrease-Key

- Select the node to be decreased,
`x`, and change its value to the new value`k`. - If the parent of
`x`,`y`, is not null and the key of parent is greater than that of the`k`then call`C`

`ut(x)`

and`C`

`ascading-`

`C`

`ut(y)`

subsequently. - If the key of
`x`is smaller than the key of min, then mark`x`as min.

### Cut

- Remove
`x`from the current position and add it to the root list. - If
`x`is marked, then mark it as false.

### Cascading-Cut

- If the parent of
`y`is not null then follow the following steps. - If
`y`is unmarked, then mark`y`. - Else, call
`Cut(y)`

and`Cascading-Cut(parent of y)`

.

## Decrease Key Example

The above operations can be understood in the examples below.

### Example: Decreasing 46 to 15.

- Decrease the value 46 to 15.

**Cut part:**Since`24 ≠ nill`

and`15 < its parent`

, cut it and add it to the root list.**Cascading-Cut part:**mark 24.

### Example: Decreasing 35 to 5

- Decrease the value 35 to 5.

- Cut part: Since
`26 ≠ nill`

and`5<its parent`

, cut it and add it to the root list.

- Cascading-Cut part: Since 26 is marked, the flow goes to
`Cut`

and`Cascading-Cut`

.

**Cut(26)**: Cut 26 and add it to the root list and mark it as false.**Cascading-Cut(24)**:

Since the 24 is also marked, again call`Cut(24)`

and`Cascading-Cut(7)`

. These operations result in the tree below. - Since
`5 < 7`

, mark 5 as min.

## Deleting a Node

This process makes use of decrease-key and extract-min operations. The following steps are followed for deleting a node.

- Let
`k`be the node to be deleted. - Apply decrease-key operation to decrease the value of
`k`to the lowest possible value (i.e. -∞). - Apply extract-min operation to remove this node.

## Python Examples

```
/* Fibonacci Heap in python */
import math
class FibonacciTree:
def __init__(self, key):
self.key = key
self.children = []
self.order = 0
def add_at_end(self, t):
self.children.append(t)
self.order = self.order + 1
class FibonacciHeap:
def __init__(self):
self.trees = []
self.least = None
self.count = 0
def insert(self, key):
new_tree = FibonacciTree(key)
self.trees.append(new_tree)
if (self.least is None or key < self.least.key):
self.least = new_tree
self.count = self.count + 1
def get_min(self):
if self.least is None:
return None
return self.least.key
def extract_min(self):
smallest = self.least
if smallest is not None:
for child in smallest.children:
self.trees.append(child)
self.trees.remove(smallest)
if self.trees == []:
self.least = None
else:
self.least = self.trees[0]
self.consolidate()
self.count = self.count - 1
return smallest.key
def consolidate(self):
aux = (floor_log2(self.count) + 1) * [None]
while self.trees != []:
x = self.trees[0]
order = x.order
self.trees.remove(x)
while aux[order] is not None:
y = aux[order]
if x.key > y.key:
x, y = y, x
x.add_at_end(y)
aux[order] = None
order = order + 1
aux[order] = x
self.least = None
for k in aux:
if k is not None:
self.trees.append(k)
if (self.least is None
or k.key < self.least.key):
self.least = k
def floor_log2(x):
return math.frexp(x)[1] - 1
fheap = FibonacciHeap()
fheap.insert(11)
fheap.insert(10)
fheap.insert(39)
fheap.insert(26)
fheap.insert(24)
print('Minimum value: {}'.format(fheap.get_min()))
print('Minimum value removed: {}'.format(fheap.extract_min()))
```

## Complexities

Decrease Key | O(1) |

Delete Node | O(log n) |

# Python Example for Beginners

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