Binary Heap

Preview

1. What is a binary heap and how is it implemented?
2. What are the main operations that can be performed on a binary heap?
3. How can a binary heap be used to efficiently sort data?
4. What is the time complexity of inserting and deleting elements in a binary heap?
5. How does the structure of a binary heap guarantee that the smallest (or largest) element is always at the top of the heap?

Explain

A Binary Heap is a type of data structure that is used to store and manage a set of ordered elements. It is called a “heap” because it resembles a pile of items where the most important items are at the top. It is a complete binary tree with a special ordering property whereby the value of each node is never less than the value of its children.

There are two types of Binary Heaps- Min Heap and Max Heap. In Min Heap, the parent node is always smaller than its child nodes while in Max Heap, the parent node is always greater than its child nodes.

Example: Suppose we have an array of numbers {10, 20, 15, 12, 40, 25, 18}. We can build a min-heap by following these steps:

1. Start with the first element (10) and create a node at the root of the heap.
2. Compare the next element (20) with the root. Since 20 is greater than 10, swap these elements.
3. Insert element 15 as the left child of 10.
4. Compare the next element (12) with its parent (20). Since 12 is less than 20, insert it as the left child of 20.
5. Compare the next element (40) with its parent (10). Since 40 is greater than 10 and 20, insert it as the right child of 20.
6. Compare the next element (25) with its parent (12). Since 25 is greater than 12, insert it as the right child of 12.
7. Compare the next element (18) with its parent (15). Since 18 is greater than 15, insert it as the right child of 15.

The resulting binary heap would look like this:

     10
   /     \
 20      15
/  \     / \

12 40 25 18

This is an example of a min heap, where the smallest element is at the root of the tree.

Keypoint

1. A binary heap is a binary tree that satisfies the heap property.
2. The heap property states that the key of any node is greater than or equal to the key of its children (if they exist).
3. Binary heaps can be represented as an array, where the root is at index 1, and the children of node i are at indices 2i and 2i+1.
4. Binary heaps are commonly used to implement priority queues.
5. The two main operations on a binary heap are insert and extract-min (or extract-max in a max heap).
6. Inserting a new element involves adding it to the bottom level of the heap and then swapping it up until the heap property is satisfied.
7. Extracting the minimum element involves removing the root and then swapping the last element in the heap to the root and sifting it down until the heap property is satisfied.
8. Binary heaps can be built from a given array of elements in O(n) time, using a bottom-up approach.
9. The height of a binary heap with n elements is logarithmic in n, making its operations very efficient.

Review

1. What is a Binary Heap?
   Answer: Binary Heap is a data structure that is a complete binary tree where all levels are completely filled, except possibly the last level.

2. What is the time complexity of finding the maximum or minimum element in a Binary Heap?
   Answer: The time complexity of finding the maximum or minimum element in a Binary Heap is O(1) as it can be easily obtained from the root node.

3. How does a Binary Heap compare to a Binary Search Tree?
   Answer: Both Binary Heap and Binary Search Tree are binary trees, but Binary Heap is used for priority queue and sorting applications, while the Binary Search Tree is used for search, insertion and deletion of dat values.

4. Can a Binary Heap have duplicate elements?
   Answer: Yes, a Binary Heap can have duplicate elements.

5. What is the process for inserting a new element into a Binary Heap?
   Answer: The process for inserting a new element into a Binary Heap is to add the new element at the end of the heap and then bubble up the element by swapping it with its parent until the heap property is maintained.