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<!DOCTYPE html>

<head>

<title>Fibonacci Heap</title>

window.dataLayer = window.dataLayer || [];

function gtag() { dataLayer.push(arguments); }

gtag('js', new Date());

gtag('config', 'G-HCERBDV76D');

</script>

</head>

<body>

<nav>

Python Tutorial

</a>

</div>

</div>

</div>

<li>

Home

</a>

</li>

<li>

Projects

</a>

</li>

<li>

Free Course

</a>

</li>

<li>

DSA

</a>

</li>

</ul>

</div>

</nav>

</div>

<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

<h1>Fibonacci Heap</h1>

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 C, C++, Java and Python.

A fibonacci heap is a data structure that consists of a collection of trees which follow min heap or max heap property. We have already discussed min heap and max heap property in the <a href="/dsa/heap-data-structure.html">Heap Data Structure</a> article. These two properties are the characteristics of the trees present on a fibonacci heap.

In a fibonacci heap, a node can have more than two children or no children at all. Also, it has more efficient heap operations than that supported by the binomial and binary heaps.

The fibonacci heap is called a fibonacci heap because the trees are constructed in a way such that a tree of order <code>n</code> has at least <code>Fn+2</code> nodes in it, where <code>Fn+2</code> is the <code>(n + 2)th</code> Fibonacci number.

<figure><img alt="Fibonacci Heap" src="https://cdn.programiz.com/sites/tutorial2program/files/fibonacci-heap.png" title="Fibonacci Heap" width="524" height="274"><figcaption>Fibonacci Heap</figcaption></figure><hr><h2 id="properties">Properties of a Fibonacci Heap</h2>

Important properties of a Fibonacci heap are:

<ol><li>It is a set of min heap-<a href="https://cs.lmu.edu/~ray/notes/orderedtrees/" target="_blank">ordered</a> trees. (i.e. The parent is always smaller than the children.)</li>

<li>A pointer is maintained at the minimum element node.</li>

<li>It consists of a set of marked nodes. (Decrease key operation)</li>

<li>The trees within a Fibonacci heap are unordered but <a href="https://mathworld.wolfram.com/RootedTree.html" target="_blank">rooted</a>.</li>

</ol><hr><h2 id="memory">Memory Representation of the Nodes in a Fibonacci Heap</h2>

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.

<ol><li>Deleting a node from the tree takes <code>O(1)</code> time.</li>

<li>The concatenation of two such lists takes <code>O(1)</code> time.</li>

</ol><figure><img alt="Fibonacci Heap Structure" src="https://cdn.programiz.com/sites/tutorial2program/files/fibonacci-heap-structure.png" title="Fibonacci Heap Structure" width="661" height="282"><figcaption>Fibonacci Heap Structure</figcaption></figure><hr><h2 id="operations">Operations on a Fibonacci Heap</h2>

<h3>Insertion</h3>

Algorithm

<pre>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</pre>

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

<ol><li>Create a new node for the element.</li>

<li>Check if the heap is empty.</li>

<li>If the heap is empty, set the new node as a root node and mark it <var>min</var>.</li>

<li>Else, insert the node into the root list and update <var>min</var>.</li>

</ol><figure><img alt="Insertion operation in fibonacci heap" src="https://cdn.programiz.com/sites/tutorial2program/files/insert-f-heap.png" title="Insertion operation in fibonacci heap" width="588" height="591"><figcaption>Insertion Example</figcaption></figure><h3>Find Min</h3>

<div class="clearfix"></div>The minimum element is always given by the <var>min</var> pointer.

<h3>Union</h3>

Union of two fibonacci heaps consists of following steps.

<ol><li>Concatenate the roots of both the heaps.</li>

<li>Update <var>min</var> by selecting a minimum key from the new root lists.</li>

</ol><figure><img alt="Union of two heaps" src="https://cdn.programiz.com/sites/tutorial2program/files/union-f-heap.png" title="Union of two heaps" width="548" height="274"><figcaption>Union of two heaps</figcaption></figure><h3>Extract Min</h3>

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:

<ol><li>Delete the min node.</li>

<li>Set the min-pointer to the next root in the root list.</li>

<li>Create an array of size equal to the maximum degree of the trees in the heap before deletion.</li>

<li>Do the following (steps 5-7) until there are no multiple roots with the same degree.</li>

<li>Map the degree of current root (min-pointer) to the degree in the array.</li>

<li>Map the degree of next root to the degree in array.</li>

<li>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).</li>

</ol>An implementation of the above steps can be understood in the example below.

<ol><li>We will perform an extract-min operation on the heap below.

<figure><img alt="Extract min" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-1.png" title="Extract min operation" width="588" height="274"><figcaption>Fibonacci Heap</figcaption></figure></li>

<li>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.

<figure><img alt="Delete the min node" src="https://www.programiz.com/sites/tutorial2program/files/extract-min-2.png" title="Extract min operation" width="588" height="274"><figcaption>Delete the min node</figcaption></figure></li>

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

<figure><img alt="Create an array" src="https://www.programiz.com/sites/tutorial2program/files/extract-min-5.png" title="Extract min operation" width="588" height="308"><figcaption>Create an array</figcaption></figure></li>

<li>Here, 23 and 7 have the same degrees, so unite them.

<figure><img alt="Unite those having the same degrees" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-6.png" title="Extract min operation" width="524" height="308"><figcaption>Unite those having the same degrees</figcaption></figure></li>

<li>Again, 7 and 17 have the same degrees, so unite them as well.

<figure><img alt="Unite those having the same degrees" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-7.png" title="Extract min operation" width="524" height="308"><figcaption>Unite those having the same degrees</figcaption></figure></li>

<li>Again 7 and 24 have the same degree, so unite them.

<figure><img alt="Unite those having the same degrees" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-8.png" title="Extract min operation" width="528" height="372"><figcaption>Unite those having the same degrees</figcaption></figure></li>

<li>Map the next nodes.

<figure><img alt="Map the remaining nodes" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-10.png" title="Extract min operation" width="528" height="384"><figcaption>Map the remaining nodes</figcaption></figure></li>

<li>Again, 52 and 21 have the same degree, so unite them

<figure><img alt="Unite those having the same degrees" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-11.png" title="Extract min operation" width="472" height="384"><figcaption>Unite those having the same degrees</figcaption></figure></li>

<li>Similarly, unite 21 and 18.

<figure><img alt="Unite those having the same degrees" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-12.png" title="Extract min operation" width="472" height="384"><figcaption>Unite those having the same degrees</figcaption></figure></li>

<li>Map the remaining root.

<figure><img alt="Map the remaining nodes" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-13.png" title="Extract min operation" width="472" height="384"><figcaption>Map the remaining nodes</figcaption></figure></li>

<li>The final heap is.

<figure><img alt="Final heap" src="https://cdn.programiz.com/sites/tutorial2program/files/extract-min-13-1.png" title="Extract min operation" width="472" height="338"><figcaption>Final fibonacci heap</figcaption></figure></li>

</ol><h3>Decreasing a Key and Deleting a Node</h3>

These are the most important operations which are discussed in <a href="/dsa/decrease-key-and-delete-node-from-a-fibonacci-heap.html">Decrease Key and Delete Node Operations</a>.

<hr><h2 id="code">Python, Java and C/C++ Examples</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"># 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()))</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs">// Operations on Fibonacci Heap in Java

// Node creation

class node {

node parent;

node left;

node right;

node child;

int degree;

boolean mark;

int key;

public node() {

this.degree = 0;

this.mark = false;

this.parent = null;

this.left = this;

this.right = this;

this.child = null;

this.key = Integer.MAX_VALUE;

}

node(int x) {

this();

this.key = x;

}

void set_parent(node x) {

this.parent = x;

}

node get_parent() {

return this.parent;

}

void set_left(node x) {

this.left = x;

}

node get_left() {

return this.left;

}

void set_right(node x) {

this.right = x;

}

node get_right() {

return this.right;

}

void set_child(node x) {

this.child = x;

}

node get_child() {

return this.child;

}

void set_degree(int x) {

this.degree = x;

}

int get_degree() {

return this.degree;

}

void set_mark(boolean m) {

this.mark = m;

}

boolean get_mark() {

return this.mark;

}

void set_key(int x) {

this.key = x;

}

int get_key() {

return this.key;

}

public class fibHeap {

node min;

int n;

boolean trace;

node found;

public boolean get_trace() {

return trace;

}

public void set_trace(boolean t) {

this.trace = t;

}

public static fibHeap create_heap() {

return new fibHeap();

}

fibHeap() {

min = null;

n = 0;

trace = false;

}

private void insert(node x) {

if (min == null) {

min = x;

x.set_left(min);

x.set_right(min);

} else {

x.set_right(min);

x.set_left(min.get_left());

min.get_left().set_right(x);

min.set_left(x);

if (x.get_key() < min.get_key())

min = x;

}

n += 1;

}

public void insert(int key) {

insert(new node(key));

}

public void display() {

display(min);

System.out.println();

}

private void display(node c) {

System.out.print("(");

if (c == null) {

System.out.print(")");

return;

} else {

node temp = c;

do {

System.out.print(temp.get_key());

node k = temp.get_child();

display(k);

System.out.print("->");

temp = temp.get_right();

} while (temp != c);

System.out.print(")");

}

public static void merge_heap(fibHeap H1, fibHeap H2, fibHeap H3) {

H3.min = H1.min;

if (H1.min != null && H2.min != null) {

node t1 = H1.min.get_left();

node t2 = H2.min.get_left();

H1.min.set_left(t2);

t1.set_right(H2.min);

H2.min.set_left(t1);

t2.set_right(H1.min);

}

if (H1.min == null || (H2.min != null && H2.min.get_key() < H1.min.get_key()))

H3.min = H2.min;

H3.n = H1.n + H2.n;

}

public int find_min() {

return this.min.get_key();

}

private void display_node(node z) {

System.out.println("right: " + ((z.get_right() == null) ? "-1" : z.get_right().get_key()));

System.out.println("left: " + ((z.get_left() == null) ? "-1" : z.get_left().get_key()));

System.out.println("child: " + ((z.get_child() == null) ? "-1" : z.get_child().get_key()));

System.out.println("degree " + z.get_degree());

}

public int extract_min() {

node z = this.min;

if (z != null) {

node c = z.get_child();

node k = c, p;

if (c != null) {

do {

p = c.get_right();

insert(c);

c.set_parent(null);

c = p;

} while (c != null && c != k);

}

z.get_left().set_right(z.get_right());

z.get_right().set_left(z.get_left());

z.set_child(null);

if (z == z.get_right())

this.min = null;

else {

this.min = z.get_right();

this.consolidate();

}

this.n -= 1;

return z.get_key();

}

return Integer.MAX_VALUE;

}

public void consolidate() {

double phi = (1 + Math.sqrt(5)) / 2;

int Dofn = (int) (Math.log(this.n) / Math.log(phi));

node[] A = new node[Dofn + 1];

for (int i = 0; i <= Dofn; ++i)

A[i] = null;

node w = min;

if (w != null) {

node check = min;

do {

node x = w;

int d = x.get_degree();

while (A[d] != null) {

node y = A[d];

if (x.get_key() > y.get_key()) {

node temp = x;

x = y;

y = temp;

w = x;

}

fib_heap_link(y, x);

check = x;

A[d] = null;

d += 1;

}

A[d] = x;

w = w.get_right();

} while (w != null && w != check);

this.min = null;

for (int i = 0; i <= Dofn; ++i) {

if (A[i] != null) {

insert(A[i]);

}

// Linking operation

private void fib_heap_link(node y, node x) {

y.get_left().set_right(y.get_right());

y.get_right().set_left(y.get_left());

node p = x.get_child();

if (p == null) {

y.set_right(y);

y.set_left(y);

} else {

y.set_right(p);

y.set_left(p.get_left());

p.get_left().set_right(y);

p.set_left(y);

}

y.set_parent(x);

x.set_child(y);

x.set_degree(x.get_degree() + 1);

y.set_mark(false);

}

// Search operation

private void find(int key, node c) {

if (found != null || c == null)

return;

else {

node temp = c;

do {

if (key == temp.get_key())

found = temp;

else {

node k = temp.get_child();

find(key, k);

temp = temp.get_right();

}

} while (temp != c && found == null);

}

public node find(int k) {

found = null;

find(k, this.min);

return found;

}

public void decrease_key(int key, int nval) {

node x = find(key);

decrease_key(x, nval);

}

// Decrease key operation

private void decrease_key(node x, int k) {

if (k > x.get_key())

return;

x.set_key(k);

node y = x.get_parent();

if (y != null && x.get_key() < y.get_key()) {

cut(x, y);

cascading_cut(y);

}

if (x.get_key() < min.get_key())

min = x;

}

// Cut operation

private void cut(node x, node y) {

x.get_right().set_left(x.get_left());

x.get_left().set_right(x.get_right());

y.set_degree(y.get_degree() - 1);

x.set_right(null);

x.set_left(null);

insert(x);

x.set_parent(null);

x.set_mark(false);

}

private void cascading_cut(node y) {

node z = y.get_parent();

if (z != null) {

if (y.get_mark() == false)

y.set_mark(true);

else {

cut(y, z);

cascading_cut(z);

}

// Delete operations

public void delete(node x) {

decrease_key(x, Integer.MIN_VALUE);

int p = extract_min();

}

public static void main(String[] args) {

fibHeap obj = create_heap();

obj.insert(7);

obj.insert(26);

obj.insert(30);

obj.insert(39);

obj.insert(10);

obj.display();