Question

Give a pseudo-code description of a nonrecursive in-place heap-sort algorithm.

Short answer

1) Build a max-heap. 2) Perform heapify. 3) Extract maximum repeatedly while maintaining the heap.

Step-by-step solution

Build a Max-Heap

Start with the given array and convert it into a max-heap. To do this, you will need to 'heapify' each sub-tree starting from the last non-leaf node up to the root. The index of the last non-leaf node is given by \(\text{floor}(n/2)-1\) where \(n\) is the length of the array.

Heapify Procedure

Define a function 'heapify' which takes three arguments: the array, the size of the heap, and the root index. In this function, determine the largest among the root, left child, and right child. If the root is not the largest, swap it with the largest and call heapify recursively on the affected sub-tree.

Convert Array to Max-Heap

Loop through all nodes from \(\text{floor}(n/2)-1\) down to 0 and call the 'heapify' function on each node. This will ensure that all sub-trees satisfy the heap property.

Extract Elements from Heap

After building the max-heap, repeatedly extract the maximum element from the heap. Swap the first element (maximum element) with the last element of the heap and reduce the heap size by one. Then, call the 'heapify' function on the root to maintain the heap property.

Repeat Extraction

Continue the process of extracting the maximum element and maintaining the heap property until the heap size is reduced to one. At this point, the array will be sorted in ascending order.

Key concepts

Max-Heap

A Max-Heap is a special kind of binary tree where the key at the root must be the largest among all the keys in the tree. This property must be recursively true for all sub-trees of the binary tree. That means each parent node must be greater than or equal to its child nodes.
In heap-sort, the first step is to build a Max-Heap from the given unsorted array. This involves ensuring that all child nodes are smaller than their parent nodes, starting from the lowest levels and moving up to the root. By the end of this process, the largest element of the array will be at the root of the Max-Heap.
The key advantage of the Max-Heap structure is that it allows us to easily extract the maximum element by simply taking the root, then restructuring the heap.

Heapify Procedure

Heapify is the process used to maintain the heap property of a node within a tree. In the context of a Max-Heap, it ensures that a parent node is greater than or equal to its children. To 'heapify', we compare the parent node with its left and right children.
If the parent is smaller than one of its children, we swap it with the larger child. We then recursively call the heapify procedure on the child node that was swapped with the parent. This process continues until the parent node is larger than both its children, or there are no children to compare.
Here is a simple outline of the heapify procedure:

• Identify the parent, left child, and right child.
• Compare the parent with children to find the largest.
• If the parent is not the largest, swap with the largest child.
• Recursively heapify the affected sub-tree if there was a swap.

Through this, we can ensure that the heap maintains its necessary properties.

In-Place Sorting

In-place sorting means sorting the array without needing extra space. The heap sort algorithm achieves this by reusing the input array for sorting. No additional array or significant extra memory is needed beyond what is used for the recursion stack when heapifying.
The in-place nature of heap sort is one of its main advantages. After building the Max-Heap, the algorithm extracts the maximum element (root) and places it at the end of the array. It then reduces the heap size and heapifies the root again to find the next largest element.
Here is a breakdown:

• Build a Max-Heap.
• Extract the maximum element and swap it with the last element.
• Reduce the heap size and heapify the root.
• Repeat until the array is sorted.

By consistently reusing the array and performing operations within it, we ensure efficient and memory-conserving sorting.