# Fibonacci Heap

#### In this tutorial, you will learn what a Fibonacci Heap is. Also, you will find working examples of different operations on a fibonacci heap in Python.

Fibonacci heap is a modified form of a binomial heap with more efficient heap operations than that supported by the binomial and binary heaps.

Unlike binary heap, a node can have more than two children.

The fibonacci heap is called 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, where `F`

`n+2`

is the `(n + 2)nd`

Fibonacci number.

## Properties of a Fibonacci Heap

Important properties of a Fibonacci heap are:

- It is a set of
**min heap-****ordered**trees. (i.e. The parent is always smaller than the children.) - A pointer is maintained at the minimum element node.
- It consists of a set of marked nodes. (Decrease key operation)
- The trees within a Fibonacci heap are unordered but rooted.

## Memory Representation of the Nodes in a Fibonacci Heap

The roots of all the trees are linked together for faster access. The child nodes of a parent node are connected to each other through a circular doubly linked list as shown below.

There are two main advantages of using a circular doubly linked list.

- Deleting a node from the tree takes
`O(1)`

time. - The concatenation of two such lists takes
`O(1)`

time.

## Operations on a Fibonacci Heap

### Insertion

Algorithm

insert(H, x) degree[x] = 0 p[x] = NIL child[x] = NIL left[x] = x right[x] = x mark[x] = FALSE concatenate the root list containing x with root list H if min[H] == NIL or key[x] < key[min[H]] then min[H] = x n[H] = n[H] + 1

Inserting a node into an already existing heap follows the steps below.

- Create a new node for the element.
- Check if the heap is empty.
- If the heap is empty, set the new node as a root node and mark it
`min`. - Else, insert the node into the root list and update
`min`.

### Find Min

The minimum element is always given by the `min` pointer.

### Union

Union of two fibonacci heaps consists of following steps.

- Concatenate the roots of both the heaps.
- Update
`min`by selecting a minimum key from the new root lists.

### Extract Min

It is the most important operation on a fibonacci heap. In this operation, the node with minimum value is removed from the heap and the tree is re-adjusted.

The following steps are followed:

- Delete the min node.
- Set the min-pointer to the next root in the root list.
- Create an array of size equal to the maximum degree of the trees in the heap before deletion.
- Do the following (steps 5-7) until there are no multiple roots with the same degree.
- Map the degree of current root (min-pointer) to the degree in the array.
- Map the degree of next root to the degree in array.
- If there are more than two mappings for the same degree, then apply union operation to those roots such that the min-heap property is maintained (i.e. the minimum is at the root).

An implementation of the above steps can be understood in the example below.

- We will perform an extract-min operation on the heap below.

- Delete the min node, add all its child nodes to the root list and set the min-pointer to the next root in the root list.

- The maximum degree in the tree is 3. Create an array of size 4 and map degree of the next roots with the array.

- Here, 23 and 7 have the same degrees, so unite them.

- Again, 7 and 17 have the same degrees, so unite them as well.

- Again 7 and 24 have the same degree, so unite them.

- Map the next nodes.

- Again, 52 and 21 have the same degree, so unite them

- Similarly, unite 21 and 18.

- Map the remaining root.

- The final heap is.

## Python Examples

```
/* Fibonacci Heap in python */
import math
/* Creating fibonacci tree */
class FibonacciTree:
def __init__(self, value):
self.value = value
self.child = []
self.order = 0
/* Adding tree at the end of the tree */
def add_at_end(self, t):
self.child.append(t)
self.order = self.order + 1
/* Creating Fibonacci heap */
class FibonacciHeap:
def __init__(self):
self.trees = []
self.least = None
self.count = 0
/* Insert a node */
def insert_node(self, value):
new_tree = FibonacciTree(value)
self.trees.append(new_tree)
if (self.least is None or value < self.least.value):
self.least = new_tree
self.count = self.count + 1
/* Get minimum value */
def get_min(self):
if self.least is None:
return None
return self.least.value
/* Extract the minimum value */
def extract_min(self):
smallest = self.least
if smallest is not None:
for child in smallest.child:
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.value
/* Consolidate the tree */
def consolidate(self):
aux = (floor_log(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.value > y.value:
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.value < self.least.value):
self.least = k
def floor_log(x):
return math.frexp(x)[1] - 1
fibonacci_heap = FibonacciHeap()
fibonacci_heap.insert_node(7)
fibonacci_heap.insert_node(3)
fibonacci_heap.insert_node(17)
fibonacci_heap.insert_node(24)
print('the minimum value of the fibonacci heap: {}'.format(fibonacci_heap.get_min()))
print('the minimum value removed: {}'.format(fibonacci_heap.extract_min()))
```

## Complexities

Insertion | O(1) |

Find Min | O(1) |

Union | O(1) |

Extract Min | O(log n) |

Decrease Key | O(1) |

Delete Node | O(log n) |

## Fibonacci Heap Applications

- To improve the asymptotic running time of Dijkstra’s algorithm.

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