# Implementation of Binomial Heap

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 :

1. insert(H, k): Inserts a key â€˜kâ€™ to Binomial Heap â€˜Hâ€™. This operation first creates a Binomial Heap with single key â€˜kâ€™, then calls union on H and the new Binomial heap.
2. 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.
3. 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 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 unionBionomialHeap(list l1,               list l2) {   // _new to another binomial heap which contain   // new heap after merging l1 & l2   list _new;   list::iterator it = l1.begin();   list::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 adjust(list _heap) {   if (_heap.size() <= 1)     return _heap;   list new_heap;   list::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 insertATreeInHeap(list _heap,               Node *tree) {   // creating a new heap i.e temp   list 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 removeMinFromTreeReturnBHeap(Node *tree) {   list 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 insert(list _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 _heap) {   list::iterator it = _heap.begin();   Node *temp = *it;   while (it != _heap.end())   {     if ((*it)->data < temp->data)       temp = *it;     it++;   }   return temp; }   list extractMin(list _heap) {   list new_heap,lo;   Node *temp;     // temp contains the pointer of minimum value   // element in heap   temp = getMin(_heap);   list::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 _heap) {   list ::iterator it;   it = _heap.begin();   while (it != _heap.end())   {     printTree(*it);     it++;   } }     // Driver program to test above functions int main() {   int ch,key;   list _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; }

## 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

## 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());     } }

Output

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

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