A Max-Heap is a complete binary tree in which the value in each internal node is greater than or equal to the values in the children of that node. Mapping the elements of a heap into an array is trivial: if a node is stored a index k, then its left child is stored at index 2k + 1 and its right child at index 2k + 2.
Examples of Max Heap :
How is Max Heap represented?
A max Heap is a Complete Binary Tree. A max heap is typically represented as an array. The root element will be at Arr[0]. 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.
Operations on Max Heap:
- getMax(): It returns the root element of Max Heap. Time Complexity of this operation is O(1).
- extractMax(): Removes the maximum element from MaxHeap. Time Complexity of this Operation is O(log n) as this operation needs to maintain the heap property (by calling heapify()) after removing the root.
- insert(): Inserting a new key takes O(log n) time. We add a new key at the end of the tree. If the new key is smaller than its parent, then we don’t need to do anything. Otherwise, we need to traverse up to fix the violated heap property.
Note: In the below implementation, we do indexing from index 1 to simplify the implementation.
Python
# Python3 implementation of Max Heapimport sysclass MaxHeap: def __init__(self, maxsize): self.maxsize = maxsize self.size = 0 self.Heap = [0] * (self.maxsize + 1) self.Heap[0] = sys.maxsize self.FRONT = 1 # Function to return the position of # parent for the node currently # at pos def parent(self, pos): return pos // 2 # Function to return the position of # the left child for the node currently # at pos def leftChild(self, pos): return 2 * pos # Function to return the position of # the right child for the node currently # at pos def rightChild(self, pos): return (2 * pos) + 1 # Function that returns true if the passed # node is a leaf node def isLeaf(self, pos): if pos >= (self.size//2) and pos <= self.size: return True return False # Function to swap two nodes of the heap def swap(self, fpos, spos): self.Heap[fpos], self.Heap[spos] = (self.Heap[spos], self.Heap[fpos]) # Function to heapify the node at pos def maxHeapify(self, pos): # If the node is a non-leaf node and smaller # than any of its child if not self.isLeaf(pos): if (self.Heap[pos] < self.Heap[self.leftChild(pos)] or self.Heap[pos] < self.Heap[self.rightChild(pos)]): # Swap with the left child and heapify # the left child if (self.Heap[self.leftChild(pos)] > self.Heap[self.rightChild(pos)]): self.swap(pos, self.leftChild(pos)) self.maxHeapify(self.leftChild(pos)) # Swap with the right child and heapify # the right child else: self.swap(pos, self.rightChild(pos)) self.maxHeapify(self.rightChild(pos)) # Function to insert a node into the heap def insert(self, element): if self.size >= self.maxsize: return self.size += 1 self.Heap[self.size] = element current = self.size while (self.Heap[current] > self.Heap[self.parent(current)]): self.swap(current, self.parent(current)) current = self.parent(current) # Function to print the contents of the heap def Print(self): for i in range(1, (self.size // 2) + 1): print("PARENT : " + str(self.Heap[i]) + "LEFT CHILD : " + str(self.Heap[2 * i]) + "RIGHT CHILD : " + str(self.Heap[2 * i + 1])) # Function to remove and return the maximum # element from the heap def extractMax(self): popped = self.Heap[self.FRONT] self.Heap[self.FRONT] = self.Heap[self.size] self.size -= 1 self.maxHeapify(self.FRONT) return popped# Driver Codeif __name__ == "__main__": print('The maxHeap is ') maxHeap = MaxHeap(15) maxHeap.insert(5) maxHeap.insert(3) maxHeap.insert(17) maxHeap.insert(10) maxHeap.insert(84) maxHeap.insert(19) maxHeap.insert(6) maxHeap.insert(22) maxHeap.insert(9) maxHeap.Print() print("The Max val is " + str(maxHeap.extractMax())) |
Output
The maxHeap is PARENT : 84LEFT CHILD : 22RIGHT CHILD : 19 PARENT : 22LEFT CHILD : 17RIGHT CHILD : 10 PARENT : 19LEFT CHILD : 5RIGHT CHILD : 6 PARENT : 17LEFT CHILD : 3RIGHT CHILD : 9 The Max val is 84
Using Library functions:
We use heapq class to implement Heap in Python. By default Min Heap is implemented by this class. But we multiply each value by -1 so that we can use it as MaxHeap.
Python3
# Python3 program to demonstrate working of heapqfrom heapq import heappop, heappush, heapify# Creating empty heapheap = []heapify(heap)# Adding items to the heap using heappush# function by multiplying them with -1heappush(heap, -1 * 10)heappush(heap, -1 * 30)heappush(heap, -1 * 20)heappush(heap, -1 * 400)# printing the value of maximum elementprint("Head value of heap : " + str(-1 * heap[0]))# printing the elements of the heapprint("The heap elements : ")for i in heap: print((-1*i), end=" ")print("\n")element = heappop(heap)# printing the elements of the heapprint("The heap elements : ")for i in heap: print(-1 * i, end = ' ') |
Output
Head value of heap : 400 The heap elements : 400 30 20 10 The heap elements : 30 10 20
Using Library functions with dunder method for Numbers, Strings, Tuples, Objects etc
We use heapq class to implement Heaps in Python. By default Min Heap is implemented by this class.
To implement MaxHeap not limiting to only numbers but any type of object(String, Tuple, Object etc) we should
- Create a Wrapper class for the item in the list.
- Override the __lt__ dunder method to give inverse result.
Following is the implementation of the method mentioned here.
Python3
"""Python3 program to implement MaxHeap Operationwith built-in module heapqfor String, Numbers, Objects"""from functools import total_orderingimport heapq# why total_ordering: https://www.geeksforgeeks.org/python-functools-total_ordering/@total_orderingclass Wrapper: def __init__(self, val): self.val = val def __lt__(self, other): return self.val > other.val def __eq__(self, other): return self.val == other.val# Working on existing list of intheap = [10, 20, 400, 30]wrapper_heap = list(map(lambda item: Wrapper(item), heap))heapq.heapify(wrapper_heap)max_item = heapq.heappop(wrapper_heap)# This will give the max valueprint(f"Top of numbers are: {max_item.val}")# Working on existing list of strheap = ["this", "code", "is", "wonderful"]wrapper_heap = list(map(lambda item: Wrapper(item), heap))heapq.heapify(wrapper_heap)print("The string heap elements in order: ")while wrapper_heap: top_item = heapq.heappop(wrapper_heap) print(top_item.val, end=" ")# This code is contributed by Supratim Samantray (super_sam) |
Output
Top of numbers are: 400 The string heap elements in order: wonderful this is code
Using internal functions used in the heapq library
This is by far the most simple and convenient way to apply max heap in python.
DISCLAIMER – In Python, there’s no strict concept of private identifiers like in C++ or Java. Python trusts developers and allows access to so-called “private” identifiers. However, since these identifiers are intended for internal use within the module, they are not officially part of the public API and may change or be removed in the future.
Following is the implementation.
Python3
from heapq import _heapify_max, _heappop_max, _siftdown_max# Implementing heappush for max_heap# Time Complexity = Θ(1 + logn)def heappush_max(max_heap, item): max_heap.append(item) _siftdown_max(max_heap, 0, len(max_heap)-1)def max_heap(arr): copy = arr.copy() # Just maintaining a copy for later use # Time Complexity = Θ(n); n = no of elements in array _heapify_max(arr) # Converts array to max_heap # Time Complexity = Θ(logk); k = no of elements in heap while len(arr) != 0: # popping items from max_heap print(_heappop_max(arr)) # ... unless its empty arr = copy # since len(arr) = 0 max_heap = [] # Time Complexity = Θ(nlogk) - since inserting item one by one for item in arr: heappush_max(max_heap, item) print("Max Heap is Ready!") # Time Complexity = Θ(logk); k = no of elements in heap while len(max_heap) != 0: # popping items from max_heap print(_heappop_max(max_heap)) # ... unless its empty arr = [6,8,9,2,1,5]max_heap(arr)# This code is contributed by Swagato Chakraborty (swagatochakraborty123) |
Output
9
8
6
5
2
1
Max Heap is Ready!
9
8
6
5
2
1
