A Min-Heap is a complete binary tree in which the value in each internal node is smaller 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 at index k, then its left child is stored at index 2k + 1 and its right child at index 2k + 2 for 0 based indexing and for 1 based indexing the left child will be at 2k and right child will be at 2k + 1.
Example of Min Heap :
5 13
/ \ / \
10 15 16 31
/ / \ / \
30 41 51 100 41
How is Min Heap represented ?
A Min Heap is a Complete Binary Tree. A Min heap is typically represented as an array. The root element will be at Arr[0]. For any ith node, i.e., Arr[i]:
- Arr[(i -1) / 2] returns its parent node.
- Arr[(2 * i) + 1] returns its left child node.
- Arr[(2 * i) + 2] returns its right child node.
Operations on Min Heap :
- getMin(): It returns the root element of Min Heap. Time Complexity of this operation is O(1).
- extractMin(): Removes the minimum element from MinHeap. 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 larger than its parent, then we don’t need to do anything. Otherwise, we need to traverse up to fix the violated heap property.
Below is the implementation of Min Heap in Python –
Python3
# Python3 implementation of Min Heap import sys class MinHeap: def __init__(self, maxsize): self.maxsize = maxsize self.size = 0 self.Heap = [0]*(self.maxsize + 1) self.Heap[0] = -1 * 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): return pos*2 > self.size # 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 minHeapify(self, pos): # If the node is a non-leaf node and greater # 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.minHeapify(self.leftChild(pos)) # Swap with the right child and heapify # the right child else: self.swap(pos, self.rightChild(pos)) self.minHeapify(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 build the min heap using # the minHeapify function def minHeap(self): for pos in range(self.size//2, 0, -1): self.minHeapify(pos) # Function to remove and return the minimum # element from the heap def remove(self): popped = self.Heap[self.FRONT] self.Heap[self.FRONT] = self.Heap[self.size] self.size-= 1 self.minHeapify(self.FRONT) return popped # Driver Codeif __name__ == "__main__": print('The minHeap is ') minHeap = MinHeap(15) minHeap.insert(5) minHeap.insert(3) minHeap.insert(17) minHeap.insert(10) minHeap.insert(84) minHeap.insert(19) minHeap.insert(6) minHeap.insert(22) minHeap.insert(9) minHeap.minHeap() minHeap.Print() print("The Min val is " + str(minHeap.remove())) |
Output :
The Min Heap is PARENT : 3 LEFT CHILD : 5 RIGHT CHILD :6 PARENT : 5 LEFT CHILD : 9 RIGHT CHILD :84 PARENT : 6 LEFT CHILD : 19 RIGHT CHILD :17 PARENT : 9 LEFT CHILD : 22 RIGHT CHILD :10 The Min val is 3
Using Library functions :
We use heapq class to implement Heaps in Python. By default Min Heap is implemented by this class.
Python3
# Python3 program to demonstrate working of heapq from heapq import heapify, heappush, heappop # Creating empty heapheap = []heapify(heap) # Adding items to the heap using heappush functionheappush(heap, 10)heappush(heap, 30)heappush(heap, 20)heappush(heap, 400) # printing the value of minimum elementprint("Head value of heap : "+str(heap[0])) # printing the elements of the heapprint("The heap elements : ")for i in heap: print(i, end = ' ')print("\n") element = heappop(heap) # printing the elements of the heapprint("The heap elements : ")for i in heap: print(i, end = ' ') |
Output :
Head value of heap : 10 The heap elements : 10 30 20 400 The heap elements : 20 30 400
