"""Binary Heap. A min heap is a complete binary tree where each node is smaller thanits children. The root, therefore, is the minimum element in the tree. The minheap uses an array to represent the data and operation. For example a min heap:
A binary heap is a special data structure that resembles a binary tree. It differs in the sense that the root of any subtree should be the smallest or the largest element.
There are two main types of heaps.
Minheap – In a minheap, the root of every subtree is the smallest element.
Maxheap – In a maxheap, the root of every subtree is the largest element.
In this article, let’s take a look at heaps and dive into programming heaps in Python.
For more background on the different types of data structures in Python, check out the following articles:
Note: Prerequisites – Make sure you have basic Python knowledge before diving into this article. It also might be a good idea to check out some linear data structures. (links are given above)
Table of Contents
Heaps: Introduction
Heaps are complete binary trees. Complete binary trees satisfy the following conditions:
All levels are filled, except the last.
All the nodes are as far left as possible.
Heaps satisfy the heap property. This means that the root of every subtree should be the greatest or smallest element in the subtree, recursively.
Applications of Heaps
Priority Queues can be implemented using heaps. The root of a heap always contains the maximum or the minimum value, based on the heap type. Therefore, a min-priority queue is implemented using a minheap. A max-priority queue is implemented using a maxheap. The element with the highest priority can be retrieved in O(1) time.
Statistics – If we want to get ordered statistics, heaps serve as a great choice. If we want the kth smallest or largest element, we can pop the heap k times to retrieve them.
Heaps are used in implementing various graph algorithms like and
Implementing a Heap
Heap Operations
A heap has the following methods:
getMax()
This operation returns the root of the maxheap.
Time Complexity - O(1).
Maxheap using List
We are going to do the list implementation of a heap. In this, the heap’s level-order traversal would be stored as an array/list.
Note - Level-Order Traversal is a recursive traversal where the root is processed first, followed by the children of the root. This is followed by the grandchildren of the root until all the nodes are processed. In the diagram above, the root node is processed first, followed by the left child, right child and so on. The final level order traversal would be: 10 4 8 50 24 5 12 18. For an overview of what a level order traversal is, check out Quora page.
In the array representation of a heap, for an element in array index i,
The Parent Node would be at position floor((i-1)/2).
The Left Child would be at position 2*i + 1.
The Right Child would be at position 2*i + 2.
Let us first define the Heap class.
This initiates a heap as a list. Now, let us define our methods.
Minheap using Heapq
We have successfully implemented a heap using a list. Now, let’s use the heapq library in Python to implement a minheap.
class MaxHeap:
def __init__(self):
# Initialize a heap using list
self.heap = []
def getParentPosition(self, i):
# The parent is located at floor((i-1)/2)
return int((i-1)/2)
def getLeftChildPosition(self, i):
# The left child is located at 2 * i + 1
return 2*i+1
def getRightChildPosition(self, i):
# The right child is located at 2 * i + 2
return 2*i+2
def hasParent(self, i):
# This function checks if the given node has a parent or not
return self.getParentPosition(i) < len(self.heap)
def hasLeftChild(self, i):
# This function checks if the given node has a left child or not
return self.getLeftChildPosition(i) < len(self.heap)
def hasRightChild(self, i):
# This function checks if the given node has a right child or not
return self.getRightChildPosition(i) < len(self.heap)
def insert(self, key):
self.heap.append(key) # Adds the key to the end of the list
self.heapify(len(self.heap) - 1) # Re-arranges the heap to maintain the heap property
def getMax(self):
return self.heap[0] # Returns the largest value in the heap in O(1) time.
def heapify(self, i):
while(self.hasParent(i) and self.heap[i] > self.heap[self.getParentPosition(i)]): # Loops until it reaches a leaf node
self.heap[i], self.heap[self.getParentPosition(i)] = self.heap[self.getParentPosition(i)], self.heap[i] # Swap the values
i = self.getParentPosition(i) # Resets the new position
def printHeap(self):
print(self.heap) # Prints the heap
import heapq
class MinHeap:
def __init__(self, minheap): # minheap is the list that we can to convert to a heap
heapq.heapify(minheap) # Use the heapify function to convert list to a heap
self.minheap = minheap
def insert(self, key):
heapq.heappush(self.minheap, key) # Insert key into the heap (heapq automatically maintains the heap property)
def getMin(self):
return self.minheap[0] # Returns the smallest element of the heap in O(1) time
def removeMin(self):
heapq.heappop(self.minheap) # The heappop function removes the smallest element in the heap
def printHeap(self):
print(self.minheap) # Prints the heap
