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Lucas Edwards
Lucas Edwards

Heap Sort Heapify Method Build Max Heap Algorithm



The latter two functions perform best for smaller values of n. For largervalues, it is more efficient to use the sorted() function. Also, whenn==1, it is more efficient to use the built-in min() and max()functions. If repeated usage of these functions is required, consider turningthe iterable into an actual heap.




Heap Sort Heapify Method Build Max Heap Algorithm



The heapsort algorithm consists of two phases: In the first phase, the array to be sorted is converted into a max heap. And in the second phase, the largest element (i.e., the one at the tree root) is removed, and a new max heap is created from the remaining elements.


The array to be sorted must first be converted into a heap. For this purpose, no new data structure is created, but the numbers are rearranged within the array so that the heap structure described above is created.


The heapify() method is called first for the last parent node. Parent nodes are 3, 7, 1, and 8. The last parent node is 8. The heapify() function checks if the children are smaller than the parent node. 4 and 6 are smaller than 8, so at this parent node, the heap condition is fulfilled, and the heapify() function is finished.


Second, heapify() is called for the penultimate node: the 1. Its children 5 and 9 are both greater than 1, so the heap condition is violated. To restore the heap condition, we now swap the larger child with the parent node, i.e., the 9 with the 1. The heapify() method is now finished again.


Since the child node we just swapped has two children itself, the heapify() method must now check if the heap condition for this child node is still valid. In this case, the 7 is greater than 4 and 6; the heap condition is fulfilled, and the heapify() function is finished.


In the following loop, the variable swapToPos iterates backward from the end of the array to its second field. In the loop body, the first element is swapped with the one at the swapToPos position, and then the heapify() method is called on the subarray up to (exclusive) the swapToPos position:


The buildHeap() method calls heapify() for each parent node, starting with the last one, and passes to this method the array, the length of the subarray representing the heap, and the position of the parent node where heapify() should start:


The heapify() method checks whether a child node is larger than the parent node. If this is the case, the parent element is swapped with the larger child element, and the process is repeated on the child node.


In the heapify() function, we walk through the tree from top to bottom. The height of a binary tree (the root not being counted) of size n is log2 n at most, i.e., if the number of elements doubles, the tree becomes only one level deeper:


We have seen above that the buildHeap() method calls heapify() for each parent node. What we have not considered so far is that the depth of the subtrees, on which heapify() is called, varies. The following graphic illustrates this (d stands for the depth of the subtrees)


Bottom-up Heapsort is a variant in which the heapify() method makes do with fewer comparisons through smart optimization. This is advantageous if, for example, we don't compare int primitives, but objects with a time-consuming compareTo() function.


Heapsort is slower than Quicksort by factor 3.6 and slower than Merge Sort by factor 2.4 for randomly distributed input data. For sorted data, heapsort is eight to nine times slower than quicksort and two times slower than Merge Sort.


Steps 2 and 3, which restore the heap property by comparing and possibly swapping a node with its parent, are called the up-heap operation (also known as bubble-up, percolate-up, sift-up, trickle-up, swim-up, heapify-up, or cascade-up).


Steps 2 and 3, which restore the heap property by comparing and possibly swapping a node with one of its children, are called the down-heap (also known as bubble-down, percolate-down, sift-down, sink-down, trickle down, heapify-down, cascade-down, extract-min or extract-max, or simply heapify) operation.


For the above algorithm to correctly re-heapify the array, no nodes besides the node at index i and its two direct children can violate the heap property. The down-heap operation (without the preceding swap) can also be used to modify the value of the root, even when an element is not being deleted.


The decrease key operation replaces the value of a node with a given value with a lower value, and the increase key operation does the same but with a higher value. This involves finding the node with the given value, changing the value, and then down-heapifying or up-heapifying to restore the heap property.


Building a heap from an array of n input elements can be done by starting with an empty heap, then successively inserting each element. This approach, called Williams' method after the inventor of binary heaps, is easily seen to run in O(n log n) time: it performs n insertions at O(log n) cost each.[a]


This implementation is used in the heapsort algorithm which reuses the space allocated to the input array to store the heap (i.e. the algorithm is done in-place). This implementation is also useful as a Priority queue. When a dynamic array is used, insertion of an unbounded number of items is possible.


The operation of merging two binary heaps takes Θ(n) for equal-sized heaps. The best you can do is (in case of array implementation) simply concatenating the two heap arrays and build a heap of the result.[13] A heap on n elements can be merged with a heap on k elements using O(log n log k) key comparisons, or, in case of a pointer-based implementation, in O(log n log k) time.[14] An algorithm for splitting a heap on n elements into two heaps on k and n-k elements, respectively, based on a new viewof heaps as an ordered collections of subheaps was presented in.[15] The algorithm requires O(log n * log n) comparisons. The view also presents a new and conceptually simple algorithm for merging heaps. When merging is a common task, a different heap implementation is recommended, such as binomial heaps, which can be merged in O(log n).


As learned earlier, there are two categories of heap data structure i.e. max-heap and min-heap. Let us understand them below but before that, we will study the heapify property to understand max-heap and min-heap.


Before moving forward with any concept, we need to learn what is heapify. So, the process of creating a heap data structure using the binary tree is called Heapify. The heapify process is used to create the Max-Heap or the Min-Heap. Let us study the Heapify using an example below:


The running time complexity of the building heap is O(n log(n)) where each call for heapify costs O(log(n)) and the cost of building heap is O(n). Therefore, the overall time complexity will be O(n log(n)).


This brings us to the end of this article where we learned about heap sort. To get a free course on data structures and algorithms, click on the banner below. Also, visit the great learning academy to see all the free courses we are providing.


In this article, we will discuss the Heapsort Algorithm. Heap sort processes the elements by creating the min-heap or max-heap using the elements of the given array. Min-heap or max-heap represents the ordering of array in which the root element represents the minimum or maximum element of the array.


Heapsort is a popular and efficient sorting algorithm. The concept of heap sort is to eliminate the elements one by one from the heap part of the list, and then insert them into the sorted part of the list.


Now let's see the working of heap sort in detail by using an example. To understand it more clearly, let's take an unsorted array and try to sort it using heap sort. It will make the explanation clearer and easier.


Next, we have to delete the root element (89) from the max heap. To delete this node, we have to swap it with the last node, i.e. (11). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, again, we have to delete the root element (81) from the max heap. To delete this node, we have to swap it with the last node, i.e. (54). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, we have to delete the root element (76) from the max heap again. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, again we have to delete the root element (54) from the max heap. To delete this node, we have to swap it with the last node, i.e. (14). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, again we have to delete the root element (22) from the max heap. To delete this node, we have to swap it with the last node, i.e. (11). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, again we have to delete the root element (14) from the max heap. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.


In the next step, again we have to delete the root element (11) from the max heap. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.


The time complexity of heap sort is O(n logn) in all three cases (best case, average case, and worst case). The height of a complete binary tree having n elements is logn. 350c69d7ab


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