Pairing Heap is like a simplified form Fibonacci Heap. It also maintains the property of min heap which is parent value is less than its child nodes value. It can be considered as a self-adjusting binomial heap.
Each node has a pointer towards the left child and left child points towards the next sibling of the child.
Example of Pairing Heap is given below:
Join or Merge in Pairing Heap
To join the two heap, first, we compare the root node of the heap if the root node of the first heap is smaller than the root node of the second heap then root node of the second heap becomes a left child of the root node of the first heap otherwise vice-versa. The time complexity of this process is O(1).
Example of Merge is given Below:
Insertion in Pairing Heap:
To insert a new node in heap, create a new node and Merge it with existing heap as explained above. Therefore, the time complexity of this function is O(1).
Example of Insertion is given below:
Deletion in Pairing Heap:
Deletion in Pairing Heap only happens at the root node. First delete links between root, left child and all the siblings of the left child. Then Merge tree subtrees that are obtained by detaching the left child and all siblings by the two pass method and delete the root node. Merge the detached subtrees from left to right in one pass and then merge the subtrees from right to left to form the new heap without violation of conditions of min-heap. This process takes O(log n) time where n is the number of nodes.
Example of Deletion is given below:
Below is the implementation of the above approach:
#include<bits/stdc++.h> using namespace std;
// Heap structure struct HeapNode {
int key;
HeapNode *leftChild;
HeapNode *nextSibling;
HeapNode():
leftChild(NULL), nextSibling(NULL) {}
// creates a new node
HeapNode( int key_, HeapNode *leftChild_, HeapNode *nextSibling_):
key(key_), leftChild(leftChild_), nextSibling(nextSibling_) {}
// Adds a child and sibling to the node
void addChild(HeapNode *node) {
if (leftChild == NULL)
leftChild = node;
else {
node->nextSibling = leftChild;
leftChild = node;
}
}
}; // Returns true if root of the tree // is null otherwise returns false bool Empty(HeapNode *node) {
return (node == NULL);
} // Function to merge two heaps HeapNode *Merge(HeapNode *A, HeapNode *B) { // If any of the two-nodes is null
// the return the not null node
if (A == NULL) return B;
if (B == NULL) return A;
// To maintain the min heap condition compare
// the nodes and node with minimum value become
// parent of the other node
if (A->key < B->key) {
A->addChild(B);
return A;
}
else {
B->addChild(A);
return B;
}
return NULL; // Unreachable
} // Returns the root value of the heap int Top(HeapNode *node) {
return node->key;
} // Function to insert the new node in the heap HeapNode *Insert(HeapNode *node, int key) {
return Merge(node, new HeapNode(key, NULL, NULL));
} // This method is used when we want to delete root node HeapNode *TwoPassMerge(HeapNode *node) { if (node == NULL || node->nextSibling == NULL)
return node;
else {
HeapNode *A, *B, *newNode;
A = node;
B = node->nextSibling;
newNode = node->nextSibling->nextSibling;
A->nextSibling = NULL;
B->nextSibling = NULL;
return Merge(Merge(A, B), TwoPassMerge(newNode));
}
return NULL; // Unreachable
} // Function to delete the root node in heap HeapNode *Delete(HeapNode *node) { return TwoPassMerge(node->leftChild);
} struct PairingHeap {
HeapNode *root;
PairingHeap():
root(NULL) {}
bool Empty( void ) {
return ::Empty(root);
}
int Top( void ) {
return ::Top(root);
}
void Insert( int key) {
root = ::Insert(root, key);
}
void Delete( void ) {
root = ::Delete(root);
}
void Join(PairingHeap other) {
root = ::Merge(root, other.root);
}
}; // Driver Code int main( void ) {
PairingHeap heap1, heap2;
heap2.Insert(5);
heap2.Insert(2);
heap2.Insert(6);
heap1.Insert(1);
heap1.Insert(3);
heap1.Insert(4);
heap1.Join(heap2);
cout << heap1.Top() << endl;
heap1.Delete();
cout << heap1.Top() << endl;
cout<< (heap1.Empty()? "True" : "False" );
return 0;
} |
import java.util.*;
// Heap structure class HeapNode {
int key;
HeapNode leftChild;
HeapNode nextSibling;
public HeapNode() {
leftChild = null ;
nextSibling = null ;
}
// creates a new node
public HeapNode( int key_, HeapNode leftChild_, HeapNode nextSibling_) {
key = key_;
leftChild = leftChild_;
nextSibling = nextSibling_;
}
// Adds a child and sibling to the node
public void addChild(HeapNode node) {
if (leftChild == null )
leftChild = node;
else {
node.nextSibling = leftChild;
leftChild = node;
}
}
} // Pairing Heap implementation class PairingHeap {
HeapNode root;
public PairingHeap() {
root = null ;
}
public boolean isEmpty() {
return root == null ;
}
public int top() {
return root.key;
}
public void insert( int key) {
root = insert(root, key);
}
public void delete() {
root = delete(root);
}
public void join(PairingHeap other) {
root = merge(root, other.root);
}
// Helper functions
private HeapNode insert(HeapNode node, int key) {
return merge(node, new HeapNode(key, null , null ));
}
private HeapNode delete(HeapNode node) {
return twoPassMerge(node.leftChild);
}
private HeapNode merge(HeapNode a, HeapNode b) {
// If any of the two nodes is null,
// return the not null node
if (a == null )
return b;
if (b == null )
return a;
// To maintain the min heap condition compare
// the nodes and node with minimum value become
// parent of the other node
if (a.key < b.key) {
a.addChild(b);
return a;
} else {
b.addChild(a);
return b;
}
}
private HeapNode twoPassMerge(HeapNode node) {
if (node == null || node.nextSibling == null )
return node;
else {
HeapNode a, b, newNode;
a = node;
b = node.nextSibling;
newNode = node.nextSibling.nextSibling;
a.nextSibling = null ;
b.nextSibling = null ;
return merge(merge(a, b), twoPassMerge(newNode));
}
}
} // Driver code public class Main {
public static void main(String[] args) {
PairingHeap heap1 = new PairingHeap();
PairingHeap heap2 = new PairingHeap();
heap2.insert( 5 );
heap2.insert( 2 );
heap2.insert( 6 );
heap1.insert( 1 );
heap1.insert( 3 );
heap1.insert( 4 );
heap1.join(heap2);
System.out.println(heap1.top());
heap1.delete();
System.out.println(heap1.top());
System.out.println(heap1.isEmpty() ? "True" : "False" );
}
} // Contributed by adityasharmadev01 |
# Heap structure class HeapNode:
# creates a new node
def __init__( self , key_ = None , leftChild_ = None , nextSibling_ = None ):
self .key = key_
self .leftChild = leftChild_
self .nextSibling = nextSibling_
# Adds a child and sibling to the node
def addChild( self , node):
if ( self .leftChild = = None ):
self .leftChild = node
else :
node.nextSibling = self .leftChild
self .leftChild = node
# Returns true if root of the tree # is None otherwise returns false def Empty(node):
return (node = = None )
# Function to merge two heaps def Merge(A, B):
# If any of the two-nodes is None
# the return the not None node
if (A = = None ):
return B
if (B = = None ):
return A
# To maintain the min heap condition compare
# the nodes and node with minimum value become
# parent of the other node
if (A.key < B.key):
A.addChild(B)
return A
B.addChild(A)
return B
# Returns the root value of the heap def Top(node):
return node.key
# Function to insert the new node in the heap def Insert(node, key):
return Merge(node, HeapNode(key,))
# This method is used when we want to delete root node def TwoPassMerge(node):
if (node = = None or node.nextSibling = = None ):
return node
A = node
B = node.nextSibling
newNode = node.nextSibling.nextSibling
A.nextSibling = None
B.nextSibling = None
return Merge(Merge(A, B), TwoPassMerge(newNode))
# Function to delete the root node in heap def Delete(node):
return TwoPassMerge(node.leftChild)
class PairingHeap:
def __init__( self ):
self .root = None
def Empty( self ):
return Empty( self .root)
def Top( self ):
return Top( self .root)
def Insert( self , key):
self .root = Insert( self .root, key)
def Delete( self ):
self .root = Delete( self .root)
def Join( self , other):
self .root = Merge( self .root, other.root)
# Driver Code if __name__ = = '__main__' :
heap1, heap2 = PairingHeap(), PairingHeap()
heap2.Insert( 5 )
heap2.Insert( 2 )
heap2.Insert( 6 )
heap1.Insert( 1 )
heap1.Insert( 3 )
heap1.Insert( 4 )
heap1.Join(heap2)
print (heap1.Top())
heap1.Delete()
print (heap1.Top())
print (heap1.Empty())
# This code is contributed by Amartya Ghosh |
// C# code for the above approach using System;
namespace GFG
{ // Heap structure
class HeapNode
{
public int Key { get ; set ; }
public HeapNode LeftChild { get ; set ; }
public HeapNode NextSibling { get ; set ; }
public HeapNode()
{
LeftChild = null ;
NextSibling = null ;
}
// Creates a new node
public HeapNode( int key, HeapNode leftChild, HeapNode nextSibling)
{
Key = key;
LeftChild = leftChild;
NextSibling = nextSibling;
}
// Adds a child and sibling to the node
public void AddChild(HeapNode node)
{
if (LeftChild == null )
LeftChild = node;
else
{
node.NextSibling = LeftChild;
LeftChild = node;
}
}
}
class PairingHeap
{
private HeapNode root;
public PairingHeap()
{
root = null ;
}
// Returns true if root of the tree
// is null otherwise returns false
public bool IsEmpty()
{
return root == null ;
}
// Returns the root value of the heap
public int Top()
{
return root.Key;
}
// Function to insert the new node in the heap
public void Insert( int key)
{
root = Merge(root, new HeapNode(key, null , null ));
}
// Function to delete the root node in heap
public void Delete()
{
root = TwoPassMerge(root.LeftChild);
}
public void Join(PairingHeap other)
{
root = Merge(root, other.root);
}
// Function to merge two heaps
private HeapNode Merge(HeapNode A, HeapNode B)
{
// If any of the two-nodes is null
// the return the not null node
if (A == null ) return B;
if (B == null ) return A;
// To maintain the min heap condition compare
// the nodes and node with minimum value become
// parent of the other node
if (A.Key < B.Key)
{
A.AddChild(B);
return A;
}
else
{
B.AddChild(A);
return B;
}
}
// This method is used when we want to delete root node
private HeapNode TwoPassMerge(HeapNode node)
{
if (node == null || node.NextSibling == null )
return node;
HeapNode A, B, newNode;
A = node;
B = node.NextSibling;
newNode = node.NextSibling.NextSibling;
A.NextSibling = null ;
B.NextSibling = null ;
return Merge(Merge(A, B), TwoPassMerge(newNode));
}
}
class Program
{
static void Main( string [] args)
{
PairingHeap heap1 = new PairingHeap();
PairingHeap heap2 = new PairingHeap();
heap2.Insert(5);
heap2.Insert(2);
heap2.Insert(6);
heap1.Insert(1);
heap1.Insert(3);
heap1.Insert(4);
heap1.Join(heap2);
Console.WriteLine(heap1.Top());
heap1.Delete();
Console.WriteLine(heap1.Top());
Console.WriteLine(heap1.IsEmpty() ? "True" : "False" );
}
}
} // This code is contributed by Abhinav Mahajan (abhinav_m22). |
// Heap structure class HeapNode { constructor(key, leftChild, nextSibling) {
this .key = key;
this .leftChild = leftChild || null ;
this .nextSibling = nextSibling || null ;
}
// Adds a child and sibling to the node
addChild(node) {
if ( this .leftChild === null ) {
this .leftChild = node;
} else {
node.nextSibling = this .leftChild;
this .leftChild = node;
}
}
} // Returns true if root of the tree // is null otherwise returns false function Empty(node) {
return node === null ;
} // Function to merge two heaps function Merge(A, B) {
// If any of the two-nodes is null
// the return the not null node
if (A === null ) {
return B;
}
if (B === null ) {
return A;
}
// To maintain the min heap condition compare
// the nodes and node with minimum value become
// parent of the other node
if (A.key < B.key) {
A.addChild(B);
return A;
} else {
B.addChild(A);
return B;
}
} // Returns the root value of the heap function Top(node) {
return node.key;
} // Function to insert the new node in the heap function Insert(node, key) {
return Merge(node, new HeapNode(key, null , null ));
} // This method is used when we want to delete root node function TwoPassMerge(node) {
if (node === null || node.nextSibling === null ) {
return node;
} else {
let A = node;
let B = node.nextSibling;
let newNode = node.nextSibling.nextSibling;
A.nextSibling = null ;
B.nextSibling = null ;
return Merge(Merge(A, B), TwoPassMerge(newNode));
}
} // Function to delete the root node in heap function Delete(node) {
return TwoPassMerge(node.leftChild);
} class PairingHeap { constructor() {
this .root = null ;
}
empty() {
return Empty( this .root);
}
top() {
return Top( this .root);
}
insert(key) {
this .root = Insert( this .root, key);
}
delete () {
this .root = Delete( this .root);
}
join(other) {
this .root = Merge( this .root, other.root);
}
} // Driver Code const heap1 = new PairingHeap();
const heap2 = new PairingHeap();
heap2.insert(5); heap2.insert(2); heap2.insert(6); heap1.insert(1); heap1.insert(3); heap1.insert(4); heap1.join(heap2); console.log(heap1.top()); heap1. delete ();
console.log(heap1.top()); console.log(heap1.empty() ? "True" : "False" );
|
1 2 False
Time Complexity:
Insertion: O(1)
Merge: O(1)
Deletion: O(logN)
Auxiliary Space: O(1).