# Binary Heap

• Difficulty Level : Medium
• Last Updated : 15 Nov, 2021

A Binary Heap is a Binary Tree with following properties.
1) It’s a complete tree (All levels are completely filled except possibly the last level and the last level has all keys as left as possible). This property of Binary Heap makes them suitable to be stored in an array.

2) A Binary Heap is either Min Heap or Max Heap. In a Min Binary Heap, the key at root must be minimum among all keys present in Binary Heap. The same property must be recursively true for all nodes in Binary Tree. Max Binary Heap is similar to MinHeap.

Examples of Min Heap:

```            10                      10
/      \               /       \
20        100          15         30
/                      /  \        /  \
30                     40    50    100   40
```

How is Binary Heap represented?
A Binary Heap is a Complete Binary Tree. A binary heap is typically represented as an array.

• The root element will be at Arr.
• Below table shows indexes of other nodes for the ith node, i.e., Arr[i]:
 Arr[(i-1)/2] Returns the parent node Arr[(2*i)+1] Returns the left child node Arr[(2*i)+2] Returns the right child node
• The traversal method use to achieve Array representation is Level Order Please refer Array Representation Of Binary Heap for details.

Applications of Heaps:
1) Heap Sort: Heap Sort uses Binary Heap to sort an array in O(nLogn) time.

2) Priority Queue: Priority queues can be efficiently implemented using Binary Heap because it supports insert(), delete() and extractmax(), decreaseKey() operations in O(logn) time. Binomoial Heap and Fibonacci Heap are variations of Binary Heap. These variations perform union also efficiently.

3) Graph Algorithms: The priority queues are especially used in Graph Algorithms like Dijkstra’s Shortest Path and Prim’s Minimum Spanning Tree.

4) Many problems can be efficiently solved using Heaps. See following for example.
a) K’th Largest Element in an array.
b) Sort an almost sorted array/
c) Merge K Sorted Arrays.

Operations on Min Heap:
1) getMini(): It returns the root element of Min Heap. Time Complexity of this operation is O(1).

2) extractMin(): Removes the minimum element from MinHeap. Time Complexity of this Operation is O(Logn) as this operation needs to maintain the heap property (by calling heapify()) after removing root.

3) decreaseKey(): Decreases value of key. The time complexity of this operation is O(Logn). If the decreases key value of a node is greater than the parent of the node, then we don’t need to do anything. Otherwise, we need to traverse up to fix the violated heap property.

4) insert(): Inserting a new key takes O(Logn) time. We add a new key at the end of the tree. IF new key is greater than its parent, then we don’t need to do anything. Otherwise, we need to traverse up to fix the violated heap property.

5) delete(): Deleting a key also takes O(Logn) time. We replace the key to be deleted with minum infinite by calling decreaseKey(). After decreaseKey(), the minus infinite value must reach root, so we call extractMin() to remove the key.

Below is the implementation of basic heap operations.

## C++

 `// A C++ program to demonstrate common Binary Heap Operations``#include``#include``using` `namespace` `std;`` ` `// Prototype of a utility function to swap two integers``void` `swap(``int` `*x, ``int` `*y);`` ` `// A class for Min Heap``class` `MinHeap``{``    ``int` `*harr; ``// pointer to array of elements in heap``    ``int` `capacity; ``// maximum possible size of min heap``    ``int` `heap_size; ``// Current number of elements in min heap``public``:``    ``// Constructor``    ``MinHeap(``int` `capacity);`` ` `    ``// to heapify a subtree with the root at given index``    ``void` `MinHeapify(``int` `);`` ` `    ``int` `parent(``int` `i) { ``return` `(i-1)/2; }`` ` `    ``// to get index of left child of node at index i``    ``int` `left(``int` `i) { ``return` `(2*i + 1); }`` ` `    ``// to get index of right child of node at index i``    ``int` `right(``int` `i) { ``return` `(2*i + 2); }`` ` `    ``// to extract the root which is the minimum element``    ``int` `extractMin();`` ` `    ``// Decreases key value of key at index i to new_val``    ``void` `decreaseKey(``int` `i, ``int` `new_val);`` ` `    ``// Returns the minimum key (key at root) from min heap``    ``int` `getMin() { ``return` `harr; }`` ` `    ``// Deletes a key stored at index i``    ``void` `deleteKey(``int` `i);`` ` `    ``// Inserts a new key 'k'``    ``void` `insertKey(``int` `k);``};`` ` `// Constructor: Builds a heap from a given array a[] of given size``MinHeap::MinHeap(``int` `cap)``{``    ``heap_size = 0;``    ``capacity = cap;``    ``harr = ``new` `int``[cap];``}`` ` `// Inserts a new key 'k'``void` `MinHeap::insertKey(``int` `k)``{``    ``if` `(heap_size == capacity)``    ``{``        ``cout << ``"\nOverflow: Could not insertKey\n"``;``        ``return``;``    ``}`` ` `    ``// First insert the new key at the end``    ``heap_size++;``    ``int` `i = heap_size - 1;``    ``harr[i] = k;`` ` `    ``// Fix the min heap property if it is violated``    ``while` `(i != 0 && harr[parent(i)] > harr[i])``    ``{``       ``swap(&harr[i], &harr[parent(i)]);``       ``i = parent(i);``    ``}``}`` ` `// Decreases value of key at index 'i' to new_val.  It is assumed that``// new_val is smaller than harr[i].``void` `MinHeap::decreaseKey(``int` `i, ``int` `new_val)``{``    ``harr[i] = new_val;``    ``while` `(i != 0 && harr[parent(i)] > harr[i])``    ``{``       ``swap(&harr[i], &harr[parent(i)]);``       ``i = parent(i);``    ``}``}`` ` `// Method to remove minimum element (or root) from min heap``int` `MinHeap::extractMin()``{``    ``if` `(heap_size <= 0)``        ``return` `INT_MAX;``    ``if` `(heap_size == 1)``    ``{``        ``heap_size--;``        ``return` `harr;``    ``}`` ` `    ``// Store the minimum value, and remove it from heap``    ``int` `root = harr;``    ``harr = harr[heap_size-1];``    ``heap_size--;``    ``MinHeapify(0);`` ` `    ``return` `root;``}`` ` ` ` `// This function deletes key at index i. It first reduced value to minus``// infinite, then calls extractMin()``void` `MinHeap::deleteKey(``int` `i)``{``    ``decreaseKey(i, INT_MIN);``    ``extractMin();``}`` ` `// A recursive method to heapify a subtree with the root at given index``// This method assumes that the subtrees are already heapified``void` `MinHeap::MinHeapify(``int` `i)``{``    ``int` `l = left(i);``    ``int` `r = right(i);``    ``int` `smallest = i;``    ``if` `(l < heap_size && harr[l] < harr[i])``        ``smallest = l;``    ``if` `(r < heap_size && harr[r] < harr[smallest])``        ``smallest = r;``    ``if` `(smallest != i)``    ``{``        ``swap(&harr[i], &harr[smallest]);``        ``MinHeapify(smallest);``    ``}``}`` ` `// A utility function to swap two elements``void` `swap(``int` `*x, ``int` `*y)``{``    ``int` `temp = *x;``    ``*x = *y;``    ``*y = temp;``}`` ` `// Driver program to test above functions``int` `main()``{``    ``MinHeap h(11);``    ``h.insertKey(3);``    ``h.insertKey(2);``    ``h.deleteKey(1);``    ``h.insertKey(15);``    ``h.insertKey(5);``    ``h.insertKey(4);``    ``h.insertKey(45);``    ``cout << h.extractMin() << ``" "``;``    ``cout << h.getMin() << ``" "``;``    ``h.decreaseKey(2, 1);``    ``cout << h.getMin();``    ``return` `0;``}`

## Python

 `# A Python program to demonstrate common binary heap operations`` ` `# Import the heap functions from python library``from` `heapq ``import` `heappush, heappop, heapify `` ` `# heappop - pop and return the smallest element from heap``# heappush - push the value item onto the heap, maintaining``#             heap invarient``# heapify - transform list into heap, in place, in linear time`` ` `# A class for Min Heap``class` `MinHeap:``     ` `    ``# Constructor to initialize a heap``    ``def` `__init__(``self``):``        ``self``.heap ``=` `[] `` ` `    ``def` `parent(``self``, i):``        ``return` `(i``-``1``)``/``2``     ` `    ``# Inserts a new key 'k'``    ``def` `insertKey(``self``, k):``        ``heappush(``self``.heap, k)           `` ` `    ``# Decrease value of key at index 'i' to new_val``    ``# It is assumed that new_val is smaller than heap[i]``    ``def` `decreaseKey(``self``, i, new_val):``        ``self``.heap[i]  ``=` `new_val ``        ``while``(i !``=` `0` `and` `self``.heap[``self``.parent(i)] > ``self``.heap[i]):``            ``# Swap heap[i] with heap[parent(i)]``            ``self``.heap[i] , ``self``.heap[``self``.parent(i)] ``=` `(``            ``self``.heap[``self``.parent(i)], ``self``.heap[i])``             ` `    ``# Method to remove minium element from min heap``    ``def` `extractMin(``self``):``        ``return` `heappop(``self``.heap)`` ` `    ``# This functon deletes key at index i. It first reduces``    ``# value to minus infinite and then calls extractMin()``    ``def` `deleteKey(``self``, i):``        ``self``.decreaseKey(i, ``float``(``"-inf"``))``        ``self``.extractMin()`` ` `    ``# Get the minimum element from the heap``    ``def` `getMin(``self``):``        ``return` `self``.heap[``0``]`` ` `# Driver pgoratm to test above function``heapObj ``=` `MinHeap()``heapObj.insertKey(``3``)``heapObj.insertKey(``2``)``heapObj.deleteKey(``1``)``heapObj.insertKey(``15``)``heapObj.insertKey(``5``)``heapObj.insertKey(``4``)``heapObj.insertKey(``45``)`` ` `print` `heapObj.extractMin(),``print` `heapObj.getMin(),``heapObj.decreaseKey(``2``, ``1``)``print` `heapObj.getMin()`` ` `# This code is contributed by Nikhil Kumar Singh(nickzuck_007)`

## C#

 `// C# program to demonstrate common ``// Binary Heap Operations - Min Heap``using` `System;`` ` `// A class for Min Heap ``class` `MinHeap{``     ` `// To store array of elements in heap``public` `int``[] heapArray{ ``get``; ``set``; }`` ` `// max size of the heap``public` `int` `capacity{ ``get``; ``set``; }`` ` `// Current number of elements in the heap``public` `int` `current_heap_size{ ``get``; ``set``; }`` ` `// Constructor ``public` `MinHeap(``int` `n)``{``    ``capacity = n;``    ``heapArray = ``new` `int``[capacity];``    ``current_heap_size = 0;``}`` ` `// Swapping using reference ``public` `static` `void` `Swap(``ref` `T lhs, ``ref` `T rhs)``{``    ``T temp = lhs;``    ``lhs = rhs;``    ``rhs = temp;``}`` ` `// Get the Parent index for the given index``public` `int` `Parent(``int` `key) ``{``    ``return` `(key - 1) / 2;``}`` ` `// Get the Left Child index for the given index``public` `int` `Left(``int` `key)``{``    ``return` `2 * key + 1;``}`` ` `// Get the Right Child index for the given index``public` `int` `Right(``int` `key)``{``    ``return` `2 * key + 2;``}`` ` `// Inserts a new key``public` `bool` `insertKey(``int` `key)``{``    ``if` `(current_heap_size == capacity)``    ``{``         ` `        ``// heap is full``        ``return` `false``;``    ``}`` ` `    ``// First insert the new key at the end ``    ``int` `i = current_heap_size;``    ``heapArray[i] = key;``    ``current_heap_size++;`` ` `    ``// Fix the min heap property if it is violated ``    ``while` `(i != 0 && heapArray[i] < ``                     ``heapArray[Parent(i)])``    ``{``        ``Swap(``ref` `heapArray[i],``             ``ref` `heapArray[Parent(i)]);``        ``i = Parent(i);``    ``}``    ``return` `true``;``}`` ` `// Decreases value of given key to new_val. ``// It is assumed that new_val is smaller ``// than heapArray[key]. ``public` `void` `decreaseKey(``int` `key, ``int` `new_val)``{``    ``heapArray[key] = new_val;`` ` `    ``while` `(key != 0 && heapArray[key] < ``                       ``heapArray[Parent(key)])``    ``{``        ``Swap(``ref` `heapArray[key], ``             ``ref` `heapArray[Parent(key)]);``        ``key = Parent(key);``    ``}``}`` ` `// Returns the minimum key (key at``// root) from min heap ``public` `int` `getMin()``{``    ``return` `heapArray;``}`` ` `// Method to remove minimum element ``// (or root) from min heap ``public` `int` `extractMin()``{``    ``if` `(current_heap_size <= 0)``    ``{``        ``return` `int``.MaxValue;``    ``}`` ` `    ``if` `(current_heap_size == 1)``    ``{``        ``current_heap_size--;``        ``return` `heapArray;``    ``}`` ` `    ``// Store the minimum value, ``    ``// and remove it from heap ``    ``int` `root = heapArray;`` ` `    ``heapArray = heapArray[current_heap_size - 1];``    ``current_heap_size--;``    ``MinHeapify(0);`` ` `    ``return` `root;``}`` ` `// This function deletes key at the ``// given index. It first reduced value ``// to minus infinite, then calls extractMin()``public` `void` `deleteKey(``int` `key)``{``    ``decreaseKey(key, ``int``.MinValue);``    ``extractMin();``}`` ` `// A recursive method to heapify a subtree ``// with the root at given index ``// This method assumes that the subtrees``// are already heapified``public` `void` `MinHeapify(``int` `key)``{``    ``int` `l = Left(key);``    ``int` `r = Right(key);`` ` `    ``int` `smallest = key;``    ``if` `(l < current_heap_size && ``        ``heapArray[l] < heapArray[smallest])``    ``{``        ``smallest = l;``    ``}``    ``if` `(r < current_heap_size && ``        ``heapArray[r] < heapArray[smallest])``    ``{``        ``smallest = r;``    ``}``     ` `    ``if` `(smallest != key)``    ``{``        ``Swap(``ref` `heapArray[key], ``             ``ref` `heapArray[smallest]);``        ``MinHeapify(smallest);``    ``}``}`` ` `// Increases value of given key to new_val.``// It is assumed that new_val is greater ``// than heapArray[key]. ``// Heapify from the given key``public` `void` `increaseKey(``int` `key, ``int` `new_val)``{``    ``heapArray[key] = new_val;``    ``MinHeapify(key);``}`` ` `// Changes value on a key``public` `void` `changeValueOnAKey(``int` `key, ``int` `new_val)``{``    ``if` `(heapArray[key] == new_val)``    ``{``        ``return``;``    ``}``    ``if` `(heapArray[key] < new_val)``    ``{``        ``increaseKey(key, new_val);``    ``} ``else``    ``{``        ``decreaseKey(key, new_val);``    ``}``}``}`` ` `static` `class` `MinHeapTest{``     ` `// Driver code``public` `static` `void` `Main(``string``[] args)``{``    ``MinHeap h = ``new` `MinHeap(11);``    ``h.insertKey(3);``    ``h.insertKey(2);``    ``h.deleteKey(1);``    ``h.insertKey(15);``    ``h.insertKey(5);``    ``h.insertKey(4);``    ``h.insertKey(45);``     ` `    ``Console.Write(h.extractMin() + ``" "``);``    ``Console.Write(h.getMin() + ``" "``);``     ` `    ``h.decreaseKey(2, 1);``    ``Console.Write(h.getMin());``}``}`` ` `// This code is contributed by ``// Dinesh Clinton Albert(dineshclinton)`

Output:

`2 4 1`