What is min heap? Create the heap for the given data structure: 96 31 27 42 34 76 61 10 4 15 6 50 3 33 45 40 80 10

What is a Min Heap?

A min heap is a complete binary tree where the value of each node is less than or equal to the values of its children, if any. The root node of a min heap contains the smallest element in the tree. Min heaps are used in several algorithms, most notably in heap sort and in the implementation of priority queues.

Here are some key properties of a min heap:

• Complete Binary Tree: A min heap is always a complete binary tree. This means that every level of the tree is fully filled, except possibly the last level, which is filled from left to right.
• Heap Property: In a min heap, for any given node i with a value A[i], the values of its children nodes A[2*i+1] and A[2*i+2] (if they exist) are greater than or equal to A[i].
• Root Element: The smallest element in a min heap is always at the root, which is A[0] in array representation.

Creating a Min Heap

To create a min heap from an arbitrary array of numbers, we can use an algorithm that builds the heap from the bottom up. We start from the last non-leaf node and heapify each node in a bottom-up manner.

Here are the steps to create a min heap from the given data:

1. Represent the data as a complete binary tree: We can represent the given data as a complete binary tree by arranging the elements in an array as they would appear in a level-order traversal of the tree.

2. Find the last non-leaf node: The last non-leaf node is the parent of the last node in the array. Since the array is zero-indexed, if n is the number of elements, the last non-leaf node is at index (n/2) - 1.

3. Heapify each node from the last non-leaf node up to the root node: To heapify a node, we compare it with its children and swap it with the smaller child if it is larger than the child. We continue this process until the node is smaller than both its children or becomes a leaf node.

4. Repeat the heapify process for all nodes up to the root node.

Let's apply these steps to the given data:

96 31 27 42 34 76 61 10 4 15 6 50 3 33 45 40 80 10

1. Represent the data as a complete binary tree:

                  96
         /                 \
       31                   27
     /    \               /    \
   42      34           76      61
  / \      / \         / \      / \
10   4   15   6      50   3   33  45
/ \
40  80

1. Find the last non-leaf node: There are 19 elements, so the last non-leaf node is at index (19/2) - 1 = 8. The value at index 8 is 4.

2. Heapify each node from the last non-leaf node up to the root node:

We start heapifying from index 8 (4), which is already a min heap since it has no children. We then move up to its parent and so on.

Let's heapify the tree in a bottom-up manner:

• Index 8: 4 (no children, already a min heap)
• Index 7: 10 (children are 40 and 80, already a min heap)
• Index 6: 61 (children are 33 and 45, swap with 33 to maintain min heap)
• Index 5: 76 (children are 50 and 3, swap with 3 to maintain min heap)
• Index 4: 34 (children are 15 and 6, swap with 6 to maintain min heap)
• Index 3: 42 (children are 10 and 4, swap with 4 to maintain min heap)
• Index 2: 27 (children are 76 and 61, already a min heap)
• Index 1: 31 (children are 42 and 34, already a min heap)
• Index 0: 96 (children are 31 and 27, swap with 27 to maintain min heap)

After heapifying index 0, we need to heapify its subtree again because the swap may have violated the min heap property in the subtree.

• Index 0: 27 (children are 31 and 96, already a min heap)

1. Repeat the heapify process for all nodes up to the root node.

After heapifying all nodes, the array representation of the min heap is:

27 31 33 42 6 76 61 10 4 15 96 50 3 34 45 40 80 10

And the corresponding complete binary tree is:

                  27
         /                 \
       31                   33
     /    \               /    \
   42      6            76      61
  / \      / \         / \      / \
10   4   15  96      50   3   34  45
/ \
40  80

This tree satisfies the min heap property, where each parent node is less than or equal to its children.