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 is 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 root.
- insert(): Inserting a new key takes O(Log n) time. We add a new key at the end of the tree. If 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 below implementation, we do indexing from index 1 to simplify the implementation.
# Python3 implementation of Max Heap import sys class 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 Code if __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 : 84 LEFT CHILD : 22 RIGHT CHILD : 19 PARENT : 22 LEFT CHILD : 17 RIGHT CHILD : 10 PARENT : 19 LEFT CHILD : 5 RIGHT CHILD : 6 PARENT : 17 LEFT CHILD : 3 RIGHT CHILD : 9 The Max val is 84
Using Library functions :
We use heapq class to implement Heaps 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 program to demonstrate working of heapq from heapq import heappop, heappush, heapify # Creating empty heap heap = [] heapify(heap) # Adding items to the heap using heappush # function by multiplying them with -1 heappush(heap, -1 * 10) heappush(heap, -1 * 30) heappush(heap, -1 * 20) heappush(heap, -1 * 400) # printing the value of maximum element print("Head value of heap : "+str(-1 * heap[0])) # printing the elements of the heap print("The heap elements : ") for i in heap: print(-1 * i, end = ' ') print("\n") element = heappop(heap) # printing the elements of the heap print("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
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