The tree involved here is always a Complete Binary Tree. And, there are two kinds of heap. First, we are asking the user to input the number of elements that are taken for sorting and then the user is allowed to enter different elements that are to be sorted. The above we have seen it for forming max heap array.
The same process is dealt with with the min-heap array formation also. As discussed above, the only difference is with the relationship between parent and child node elements. Though there are many sorting techniques, heap sort is considered one of the better sorting technique due to its time and space complexity.
The time complexity for all best, average, and worst case is O nlogn , where worst-case complexity is better than worst-case complexity of Quicksort and space complexity is O 1. This is a guide to Heap Sort in C. Here we discuss the logic and the Steps for Heap Sort with the sample code and output along with pictorial representations. You may also have a look at the following articles to learn more —. Submit Next Question.
By signing up, you agree to our Terms of Use and Privacy Policy. Forgot Password? This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. A binary tree is said to follow a heap data structure if. Starting from a complete binary tree, we can modify it to become a Max-Heap by running a function called heapify on all the non-leaf elements of the heap. Since heapify uses recursion, it can be difficult to grasp.
So let's first think about how you would heapify a tree with just three elements. The example above shows two scenarios - one in which the root is the largest element and we don't need to do anything. And another in which the root had a larger element as a child and we needed to swap to maintain max-heap property.
If you're worked with recursive algorithms before, you've probably identified that this must be the base case. To maintain the max-heap property for the entire tree, we will have to keep pushing 2 downwards until it reaches its correct position. Thus, to maintain the max-heap property in a tree where both sub-trees are max-heaps, we need to run heapify on the root element repeatedly until it is larger than its children or it becomes a leaf node.
This function works for both the base case and for a tree of any size. We can thus move the root element to the correct position to maintain the max-heap status for any tree size as long as the sub-trees are max-heaps. To build a max-heap from any tree, we can thus start heapifying each sub-tree from the bottom up and end up with a max-heap after the function is applied to all the elements including the root element. All other nodes after that are leaf-nodes and thus don't need to be heapified.
As shown in the above diagram, we start by heapifying the lowest smallest trees and gradually move up until we reach the root element. If you've understood everything till here, congratulations, you are on your way to mastering the Heap sort.
Heap Sort has O nlog n time complexities for all the cases best case, average case, and worst case. Let us understand the reason why. The height of a complete binary tree containing n elements is log n. As we have seen earlier, to fully heapify an element whose subtrees are already max-heaps, we need to keep comparing the element with its left and right children and pushing it downwards until it reaches a point where both its children are smaller than it.
In the worst case scenario, we will need to move an element from the root to the leaf node making a multiple of log n comparisons and swaps. During the sorting step, we exchange the root element with the last element and heapify the root element.
For each element, this again takes log n worst time because we might have to bring the element all the way from the root to the leaf. Also it performs sorting in O 1 space complexity. Compared with Quick Sort, it has a better worst case O nlog n. But in other cases, Quick Sort is fast. Introsort is an alternative to heapsort that combines quicksort and heapsort to retain advantages of both: worst case speed of heapsort and average speed of quicksort.
Systems concerned with security and embedded systems such as Linux Kernel use Heap Sort because of the O n log n upper bound on Heapsort's running time and constant O 1 upper bound on its auxiliary storage. Although Heap Sort has O n log n time complexity even for the worst case, it doesn't have more applications compared to other sorting algorithms like Quick Sort, Merge Sort. However, its underlying data structure, heap, can be efficiently used if we want to extract the smallest or largest from the list of items without the overhead of keeping the remaining items in the sorted order.
For e. Course Index Explore Programiz. Start Learning DSA. Popular Tutorials Quicksort Algorithm. Merge Sort Algorithm. Linked List Data Structure.
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