Implementation of Binomial Heap

Last Updated : 04 Apr, 2024

In previous article, we have discussed about the concepts related to Binomial heap.

Examples Binomial Heap:

12------------10--------------------20
             /  \                 /  | \
           15    50             70  50  40
           |                  / |    |     
           30               80  85  65 
                            |
                           100
A Binomial Heap with 13 nodes. It is a collection of 3 
Binomial Trees of orders 0, 2 and 3 from left to right. 

    10--------------------20
   /  \                 /  | \
 15    50             70  50  40
 |                  / |    |     
 30               80  85  65 
                  |
                 100

In this article, implementation of Binomial Heap is discussed. Following functions implemented :

insert(H, k): Inserts a key ‘k’ to Binomial Heap ‘H’. This operation first creates a Binomial Heap with single key ‘k’, then calls union on H and the new Binomial heap.
getMin(H): A simple way to getMin() is to traverse the list of root of Binomial Trees and return the minimum key. This implementation requires O(Logn) time. It can be optimized to O(1) by maintaining a pointer to minimum key root.
extractMin(H): This operation also uses union(). We first call getMin() to find the minimum key Binomial Tree, then we remove the node and create a new Binomial Heap by connecting all subtrees of the removed minimum node. Finally we call union() on H and the newly created Binomial Heap. This operation requires O(Logn) time.

Implementation:

CPP

// C++ program to implement different operations
// on Binomial Heap
#include<bits/stdc++.h>
using namespace std;

// A Binomial Tree node.
struct Node
{
  int data, degree;
  Node *child, *sibling, *parent;
};

Node* newNode(int key)
{
  Node *temp = new Node;
  temp->data = key;
  temp->degree = 0;
  temp->child = temp->parent = temp->sibling = NULL;
  return temp;
}

// This function merge two Binomial Trees.
Node* mergeBinomialTrees(Node *b1, Node *b2)
{
  // Make sure b1 is smaller
  if (b1->data > b2->data)
    swap(b1, b2);

  // We basically make larger valued tree
  // a child of smaller valued tree
  b2->parent = b1;
  b2->sibling = b1->child;
  b1->child = b2;
  b1->degree++;

  return b1;
}

// This function perform union operation on two
// binomial heap i.e. l1 & l2
list<Node*> unionBionomialHeap(list<Node*> l1,
              list<Node*> l2)
{
  // _new to another binomial heap which contain
  // new heap after merging l1 & l2
  list<Node*> _new;
  list<Node*>::iterator it = l1.begin();
  list<Node*>::iterator ot = l2.begin();
  while (it!=l1.end() && ot!=l2.end())
  {
    // if D(l1) <= D(l2)
    if((*it)->degree <= (*ot)->degree)
    {
      _new.push_back(*it);
      it++;
    }
    // if D(l1) > D(l2)
    else
    {
      _new.push_back(*ot);
      ot++;
    }
  }

  // if there remains some elements in l1
  // binomial heap
  while (it != l1.end())
  {
    _new.push_back(*it);
    it++;
  }

  // if there remains some elements in l2
  // binomial heap
  while (ot!=l2.end())
  {
    _new.push_back(*ot);
    ot++;
  }
  return _new;
}

// adjust function rearranges the heap so that
// heap is in increasing order of degree and
// no two binomial trees have same degree in this heap
list<Node*> adjust(list<Node*> _heap)
{
  if (_heap.size() <= 1)
    return _heap;
  list<Node*> new_heap;
  list<Node*>::iterator it1,it2,it3;
  it1 = it2 = it3 = _heap.begin();

  if (_heap.size() == 2)
  {
    it2 = it1;
    it2++;
    it3 = _heap.end();
  }
  else
  {
    it2++;
    it3=it2;
    it3++;
  }
  while (it1 != _heap.end())
  {
    // if only one element remains to be processed
    if (it2 == _heap.end())
      it1++;

    // If D(it1) < D(it2) i.e. merging of Binomial
    // Tree pointed by it1 & it2 is not possible
    // then move next in heap
    else if ((*it1)->degree < (*it2)->degree)
    {
      it1++;
      it2++;
      if(it3!=_heap.end())
        it3++;
    }

    // if D(it1),D(it2) & D(it3) are same i.e.
    // degree of three consecutive Binomial Tree are same
    // in heap
    else if (it3!=_heap.end() &&
        (*it1)->degree == (*it2)->degree &&
        (*it1)->degree == (*it3)->degree)
    {
      it1++;
      it2++;
      it3++;
    }

    // if degree of two Binomial Tree are same in heap
    else if ((*it1)->degree == (*it2)->degree)
    {
      Node *temp;
      *it1 = mergeBinomialTrees(*it1,*it2);
      it2 = _heap.erase(it2);
      if(it3 != _heap.end())
        it3++;
    }
  }
  return _heap;
}

// inserting a Binomial Tree into binomial heap
list<Node*> insertATreeInHeap(list<Node*> _heap,
              Node *tree)
{
  // creating a new heap i.e temp
  list<Node*> temp;

  // inserting Binomial Tree into heap
  temp.push_back(tree);

  // perform union operation to finally insert
  // Binomial Tree in original heap
  temp = unionBionomialHeap(_heap,temp);

  return adjust(temp);
}

// removing minimum key element from binomial heap
// this function take Binomial Tree as input and return
// binomial heap after
// removing head of that tree i.e. minimum element
list<Node*> removeMinFromTreeReturnBHeap(Node *tree)
{
  list<Node*> heap;
  Node *temp = tree->child;
  Node *lo;

  // making a binomial heap from Binomial Tree
  while (temp)
  {
    lo = temp;
    temp = temp->sibling;
    lo->sibling = NULL;
    heap.push_front(lo);
  }
  return heap;
}

// inserting a key into the binomial heap
list<Node*> insert(list<Node*> _head, int key)
{
  Node *temp = newNode(key);
  return insertATreeInHeap(_head,temp);
}

// return pointer of minimum value Node
// present in the binomial heap
Node* getMin(list<Node*> _heap)
{
  list<Node*>::iterator it = _heap.begin();
  Node *temp = *it;
  while (it != _heap.end())
  {
    if ((*it)->data < temp->data)
      temp = *it;
    it++;
  }
  return temp;
}

list<Node*> extractMin(list<Node*> _heap)
{
  list<Node*> new_heap,lo;
  Node *temp;

  // temp contains the pointer of minimum value
  // element in heap
  temp = getMin(_heap);
  list<Node*>::iterator it;
  it = _heap.begin();
  while (it != _heap.end())
  {
    if (*it != temp)
    {
      // inserting all Binomial Tree into new
      // binomial heap except the Binomial Tree
      // contains minimum element
      new_heap.push_back(*it);
    }
    it++;
  }
  lo = removeMinFromTreeReturnBHeap(temp);
  new_heap = unionBionomialHeap(new_heap,lo);
  new_heap = adjust(new_heap);
  return new_heap;
}

// print function for Binomial Tree
void printTree(Node *h)
{
  while (h)
  {
    cout << h->data << " ";
    printTree(h->child);
    h = h->sibling;
  }
}

// print function for binomial heap
void printHeap(list<Node*> _heap)
{
  list<Node*> ::iterator it;
  it = _heap.begin();
  while (it != _heap.end())
  {
    printTree(*it);
    it++;
  }
}


// Driver program to test above functions
int main()
{
  int ch,key;
  list<Node*> _heap;

  // Insert data in the heap
  _heap = insert(_heap,10);
  _heap = insert(_heap,20);
  _heap = insert(_heap,30);

  cout << "Heap elements after insertion:\n";
  printHeap(_heap);

  Node *temp = getMin(_heap);
  cout << "\nMinimum element of heap "
    << temp->data << "\n";

  // Delete minimum element of heap
  _heap = extractMin(_heap);
  cout << "Heap after deletion of minimum element\n";
  printHeap(_heap);

  return 0;
}

Java

// Java Program to Implement Binomial Heap

// Importing required classes
import java.io.*;

// Class 1
// BinomialHeapNode
class BinomialHeapNode {

    int key, degree;
    BinomialHeapNode parent;
    BinomialHeapNode sibling;
    BinomialHeapNode child;

    // Constructor of this class
    public BinomialHeapNode(int k)
    {

        key = k;
        degree = 0;
        parent = null;
        sibling = null;
        child = null;
    }

    // Method 1
    // To reverse
    public BinomialHeapNode reverse(BinomialHeapNode sibl)
    {
        BinomialHeapNode ret;
        if (sibling != null)
            ret = sibling.reverse(this);
        else
            ret = this;
        sibling = sibl;
        return ret;
    }

    // Method 2
    // To find minimum node
    public BinomialHeapNode findMinNode()
    {

        // this keyword refers to current instance itself
        BinomialHeapNode x = this, y = this;
        int min = x.key;

        while (x != null) {
            if (x.key < min) {
                y = x;
                min = x.key;
            }

            x = x.sibling;
        }

        return y;
    }

    // Method 3
    // To find node with key value
    public BinomialHeapNode findANodeWithKey(int value)
    {

        BinomialHeapNode temp = this, node = null;

        while (temp != null) {
            if (temp.key == value) {
                node = temp;
                break;
            }

            if (temp.child == null)
                temp = temp.sibling;

            else {
                node = temp.child.findANodeWithKey(value);
                if (node == null)
                    temp = temp.sibling;
                else
                    break;
            }
        }

        return node;
    }

    // Method 4
    // To get the size
    public int getSize()
    {
        return (
            1 + ((child == null) ? 0 : child.getSize())
            + ((sibling == null) ? 0 : sibling.getSize()));
    }
}

// Class 2
// BinomialHeap
class BinomialHeap {

    // Member variables of this class
    private BinomialHeapNode Nodes;
    private int size;

    // Constructor of this class
    public BinomialHeap()
    {
        Nodes = null;
        size = 0;
    }

    // Checking if heap is empty
    public boolean isEmpty() { return Nodes == null; }

    // Method 1
    // To get the size
    public int getSize() { return size; }

    // Method 2
    // Clear heap
    public void makeEmpty()
    {
        Nodes = null;
        size = 0;
    }

    // Method 3
    // To insert
    public void insert(int value)
    {

        if (value > 0) {
            BinomialHeapNode temp
                = new BinomialHeapNode(value);
            if (Nodes == null) {
                Nodes = temp;
                size = 1;
            }
            else {
                unionNodes(temp);
                size++;
            }
        }
    }

    // Method 4
    // To unite two binomial heaps
    private void merge(BinomialHeapNode binHeap)
    {
        BinomialHeapNode temp1 = Nodes, temp2 = binHeap;

        while ((temp1 != null) && (temp2 != null)) {

            if (temp1.degree == temp2.degree) {

                BinomialHeapNode tmp = temp2;
                temp2 = temp2.sibling;
                tmp.sibling = temp1.sibling;
                temp1.sibling = tmp;
                temp1 = tmp.sibling;
            }

            else {

                if (temp1.degree < temp2.degree) {

                    if ((temp1.sibling == null)
                        || (temp1.sibling.degree
                            > temp2.degree)) {
                        BinomialHeapNode tmp = temp2;
                        temp2 = temp2.sibling;
                        tmp.sibling = temp1.sibling;
                        temp1.sibling = tmp;
                        temp1 = tmp.sibling;
                    }

                    else {
                        temp1 = temp1.sibling;
                    }
                }

                else {
                    BinomialHeapNode tmp = temp1;
                    temp1 = temp2;
                    temp2 = temp2.sibling;
                    temp1.sibling = tmp;

                    if (tmp == Nodes) {
                        Nodes = temp1;
                    }

                    else {
                    }
                }
            }
        }

        if (temp1 == null) {
            temp1 = Nodes;

            while (temp1.sibling != null) {
                temp1 = temp1.sibling;
            }
            temp1.sibling = temp2;
        }

        else {
        }
    }

    // Method 5
    // For union of nodes
    private void unionNodes(BinomialHeapNode binHeap)
    {
        merge(binHeap);

        BinomialHeapNode prevTemp = null, temp = Nodes,
                        nextTemp = Nodes.sibling;

        while (nextTemp != null) {

            if ((temp.degree != nextTemp.degree)
                || ((nextTemp.sibling != null)
                    && (nextTemp.sibling.degree
                        == temp.degree))) {
                prevTemp = temp;
                temp = nextTemp;
            }

            else {

                if (temp.key <= nextTemp.key) {
                    temp.sibling = nextTemp.sibling;
                    nextTemp.parent = temp;
                    nextTemp.sibling = temp.child;
                    temp.child = nextTemp;
                    temp.degree++;
                }

                else {

                    if (prevTemp == null) {
                        Nodes = nextTemp;
                    }

                    else {
                        prevTemp.sibling = nextTemp;
                    }

                    temp.parent = nextTemp;
                    temp.sibling = nextTemp.child;
                    nextTemp.child = temp;
                    nextTemp.degree++;
                    temp = nextTemp;
                }
            }
            nextTemp = temp.sibling;
        }
    }

    // Method 6
    // To return minimum key
    public int findMinimum()
    {
        return Nodes.findMinNode().key;
    }

    // Method 7
    // To delete a particular element */
    public void delete(int value)
    {

        if ((Nodes != null)
            && (Nodes.findANodeWithKey(value) != null)) {
            decreaseKeyValue(value, findMinimum() - 1);
            extractMin();
        }
    }

    // Method 8
    // To decrease key with a given value */
    public void decreaseKeyValue(int old_value,
                                int new_value)
    {
        BinomialHeapNode temp
            = Nodes.findANodeWithKey(old_value);
        if (temp == null)
            return;
        temp.key = new_value;
        BinomialHeapNode tempParent = temp.parent;

        while ((tempParent != null)
            && (temp.key < tempParent.key)) {
            int z = temp.key;
            temp.key = tempParent.key;
            tempParent.key = z;

            temp = tempParent;
            tempParent = tempParent.parent;
        }
    }

    // Method 9
    // To extract the node with the minimum key
    public int extractMin()
    {
        if (Nodes == null)
            return -1;

        BinomialHeapNode temp = Nodes, prevTemp = null;
        BinomialHeapNode minNode = Nodes.findMinNode();

        while (temp.key != minNode.key) {
            prevTemp = temp;
            temp = temp.sibling;
        }

        if (prevTemp == null) {
            Nodes = temp.sibling;
        }
        else {
            prevTemp.sibling = temp.sibling;
        }

        temp = temp.child;
        BinomialHeapNode fakeNode = temp;

        while (temp != null) {
            temp.parent = null;
            temp = temp.sibling;
        }

        if ((Nodes == null) && (fakeNode == null)) {
            size = 0;
        }
        else {
            if ((Nodes == null) && (fakeNode != null)) {
                Nodes = fakeNode.reverse(null);
                size = Nodes.getSize();
            }
            else {
                if ((Nodes != null) && (fakeNode == null)) {
                    size = Nodes.getSize();
                }
                else {
                    unionNodes(fakeNode.reverse(null));
                    size = Nodes.getSize();
                }
            }
        }

        return minNode.key;
    }

    // Method 10
    // To display heap
    public void displayHeap()
    {
        System.out.print("\nHeap : ");
        displayHeap(Nodes);
        System.out.println("\n");
    }

    private void displayHeap(BinomialHeapNode r)
    {
        if (r != null) {
            displayHeap(r.child);
            System.out.print(r.key + " ");
            displayHeap(r.sibling);
        }
    }
}

// Class 3
// Main class
public class GFG {
    public static void main(String[] args)
    {

        // Make object of BinomialHeap
        BinomialHeap binHeap = new BinomialHeap();

        // Inserting in the binomial heap
        // Custom input integer values
        binHeap.insert(12);
        binHeap.insert(8);
        binHeap.insert(5);
        binHeap.insert(15);
        binHeap.insert(7);
        binHeap.insert(2);
        binHeap.insert(9);

        // Size of binomial heap
        System.out.println("Size of the binomial heap is "
                        + binHeap.getSize());

        // Displaying the binomial heap
        binHeap.displayHeap();

        // Deletion in binomial heap
        binHeap.delete(15);
        binHeap.delete(8);

        // Size of binomial heap
        System.out.println("Size of the binomial heap is "
                        + binHeap.getSize());

        // Displaying the binomial heap
        binHeap.displayHeap();

        // Making the heap empty
        binHeap.makeEmpty();

        // checking if heap is empty
        System.out.println(binHeap.isEmpty());
    }
}

Python3

"""
Min Heap Implementation in Python
"""
class MinHeap:
    def __init__(self):
        """
        On this implementation the heap list is initialized with a value
        """
        self.heap_list = [0]
        self.current_size = 0
 
    def sift_up(self, i):
        """
        Moves the value up in the tree to maintain the heap property.
        """
        # While the element is not the root or the left element
        Stop = False
        while (i // 2 > 0) and Stop == False:
            # If the element is less than its parent swap the elements
            if self.heap_list[i] < self.heap_list[i // 2]:
                self.heap_list[i], self.heap_list[i // 2] = self.heap_list[i // 2], self.heap_list[i]
            else:
                Stop = True
            # Move the index to the parent to keep the properties
            i = i // 2
 
    def insert(self, k):
        """
        Inserts a value into the heap
        """
        # Append the element to the heap
        self.heap_list.append(k)
        # Increase the size of the heap.
        self.current_size += 1
        # Move the element to its position from bottom to the top
        self.sift_up(self.current_size)
 
    def sift_down(self, i):
        # if the current node has at least one child
        while (i * 2) <= self.current_size:
            # Get the index of the min child of the current node
            mc = self.min_child(i)
            # Swap the values of the current element is greater than its min child
            if self.heap_list[i] > self.heap_list[mc]:
                self.heap_list[i], self.heap_list[mc] = self.heap_list[mc], self.heap_list[i]
            i = mc
 
    def min_child(self, i):
        # If the current node has only one child, return the index of the unique child
        if (i * 2)+1 > self.current_size:
            return i * 2
        else:
            # Herein the current node has two children
            # Return the index of the min child according to their values
            if self.heap_list[i*2] < self.heap_list[(i*2)+1]:
                return i * 2
            else:
                return (i * 2) + 1
 
    def delete_min(self):
        # Equal to 1 since the heap list was initialized with a value
        if len(self.heap_list) == 1:
            return 'Empty heap'
 
        # Get root of the heap (The min value of the heap)
        root = self.heap_list[1]
 
        # Move the last value of the heap to the root
        self.heap_list[1] = self.heap_list[self.current_size]
 
        # Pop the last value since a copy was set on the root
        *self.heap_list, _ = self.heap_list
 
        # Decrease the size of the heap
        self.current_size -= 1
 
        # Move down the root (value at index 1) to keep the heap property
        self.sift_down(1)
 
        # Return the min value of the heap
        return root
"""
Driver program
"""
# Same tree as above example.
my_heap = MinHeap()
my_heap.insert(5)
my_heap.insert(6)
my_heap.insert(7)
my_heap.insert(9)
my_heap.insert(13)
my_heap.insert(11)
my_heap.insert(10)

print(my_heap.delete_min()) # removing min node i.e 5

JavaScript

// JavaScript program to implement different operations
// on Binomial Heap

// A Binomial Tree node.
class Node {
  constructor(data) {
    this.data = data;
    this.degree = 0;
    this.child = null;
    this.sibling = null;
    this.parent = null;
  }
}

// This function merges two Binomial Trees.
function mergeBinomialTrees(b1, b2) {
  // Make sure b1 is smaller
  if (b1.data > b2.data) {
    let temp = b1;
    b1 = b2;
    b2 = temp;
  }

  // We basically make larger valued tree
  // a child of smaller valued tree
  b2.parent = b1;
  b2.sibling = b1.child;
  b1.child = b2;
  b1.degree++;

  return b1;
}

// This function performs union operation on two
// binomial heaps i.e. l1 & l2
function unionBionomialHeap(l1, l2) {
  // _new to another binomial heap which contain
  // new heap after merging l1 & l2
  let _new = [];
  let it = 0;
  let ot = 0;
  while (it < l1.length && ot < l2.length) {
    // if D(l1) <= D(l2)
    if(l1[it].degree <= l2[ot].degree) {
      _new.push(l1[it]);
      it++;
    }
    // if D(l1) > D(l2)
    else {
      _new.push(l2[ot]);
      ot++;
    }
  }

  // if there remains some elements in l1
  // binomial heap
  while (it < l1.length) {
    _new.push(l1[it]);
    it++;
  }

  // if there remains some elements in l2
  // binomial heap
  while (ot < l2.length) {
    _new.push(l2[ot]);
    ot++;
  }
  return _new;
}

// adjust function rearranges the heap so that
// heap is in increasing order of degree and
// no two binomial trees have same degree in this heap
function adjust(_heap) {
  if (_heap.length <= 1)
    return _heap;
  let new_heap = [];
  let it1 = 0;
  let it2 = 1;
  let it3 = 2;
  while (it1 < _heap.length) {
    // if only one element remains to be processed
    if (it2 >= _heap.length)
      it1++;

    // If D(it1) < D(it2) i.e. merging of Binomial
    // Tree pointed by it1 & it2 is not possible
    // then move next in heap
    else if (_heap[it1].degree < _heap[it2].degree) {
      it1++;
      it2++;
      if(it3 < _heap.length)
        it3++;
    }

    // if D(it1),D(it2) & D(it3) are same i.e.
    // degree of three consecutive Binomial Tree are same
    // in heap
    else if (it3 < _heap.length &&
        _heap[it1].degree == _heap[it2].degree &&
        _heap[it1].degree == _heap[it3].degree) {
      it1++;
      it2++;
      it3++;
    }

    // if degree of two Binomial Tree are same in heap
    else if (_heap[it1].degree == _heap[it2].degree) {
      _heap[it1] = mergeBinomialTrees(_heap[it1], _heap[it2]);
      _heap.splice(it2, 1);
      if(it3 < _heap.length)
        it3++;
    }
  }
  return _heap;
}

// inserting a Binomial Tree into binomial heap
function insertATreeInHeap(_heap, tree) {
  // creating a new heap i.e temp
  let temp = [];

  // inserting Binomial Tree into heap
  temp.push(tree);

  // perform union operation to finally insert
  // Binomial Tree in original heap
  temp = unionBionomialHeap(_heap, temp);

  return adjust(temp);
}

// removing minimum key element from binomial heap
// this function takes Binomial Tree as input and returns
// binomial heap after
// removing head of that tree i.e. minimum element
function removeMinFromTreeReturnBHeap(tree) {
  let heap = [];
  let temp = tree.child;
  let lo;

  // making a binomial heap from Binomial Tree
  while (temp) {
    lo = temp;
    temp = temp.sibling;
    lo.sibling = null;
    heap.unshift(lo);
  }
  return heap;
}

// inserting a key into the binomial heap
function insert(_head, key) {
  let temp = new Node(key);
  return insertATreeInHeap(_head, temp);
}

// return pointer of minimum value Node
// present in the binomial heap
function getMin(_heap) {
  let it = 0;
  let temp = _heap[0];
  while (it < _heap.length) {
    if (_heap[it].data < temp.data)
      temp = _heap[it];
    it++;
  }
  return temp;
}

function extractMin(_heap) {
  let new_heap = [];
  let lo;
  let temp;

  // temp contains the pointer of minimum value
  // element in heap
  temp = getMin(_heap);
  let it = 0;
  while (it < _heap.length) {
    if (_heap[it] != temp) {
      // inserting all Binomial Tree into new
      // binomial heap except the Binomial Tree
      // contains minimum element
      new_heap.push(_heap[it]);
    }
    it++;
  }
  lo = removeMinFromTreeReturnBHeap(temp);
  new_heap = unionBionomialHeap(new_heap, lo);
  new_heap = adjust(new_heap);
  return new_heap;
}

// print function for Binomial Tree
function printTree(h) {
  while (h) {
    console.log(h.data + " ");
    printTree(h.child);
    h = h.sibling;
  }
}

// print function for binomial heap
function printHeap(_heap) {
  for(let i = 0; i < _heap.length; i++) {
    printTree(_heap[i]);
  }
}

// Driver program to test above functions
function main() {
  let _heap = [];

  // Insert data in the heap
  _heap = insert(_heap, 10);
  _heap = insert(_heap, 20);
  _heap = insert(_heap, 30);

  console.log("Heap elements after insertion:");
  printHeap(_heap);

  let temp = getMin(_heap);
  console.log("\nMinimum element of heap " + temp.data);

  // Delete minimum element of heap
  _heap = extractMin(_heap);
  console.log("Heap after deletion of minimum element");
  printHeap(_heap);
}

main();

Output

Heap elements after insertion:
30 10 20 
Minimum element of heap 10
Heap after deletion of minimum element
20 30

Implementation of Binomial Heap | Set - 2 (delete() and decreseKey())

Sahil Chhabra (akku) and Arun Mittal

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Implementation of Binomial Heap

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