# Heap Sort Algorithm

May 20, 2023

Heap sort is a comparison-based sorting algorithm that efficiently sorts an array by converting it into a binary heap, and then continually removing the root element and inserting it into a sorted array. Heap sort is an in-place algorithm that has an average case time complexity of O(n log n) and a worst case time complexity of O(n log n), which makes it very useful for sorting large quantities of data.

Heap sort is commonly used in computer science as a general-purpose comparison-based sorting algorithm, and it is particularly useful for applications that require stable sorting or sorting of data that is too large to fit into memory.

## Brief History and Development

The concept of binary heaps was first introduced by J. W. J. Williams in 1964, but the heap sort algorithm was not developed until 1968 by Robert W. Floyd. Floyd’s algorithm is a straightforward implementation of the binary heap data structure, which has since been improved upon by various computer scientists, including J. W. J. Williams and J. B. Kruskal.

Heap sort was popularized in the 1970s and 1980s, when it was widely used in computer science textbooks as a way to teach basic data structures and algorithms. Today, heap sort is still used in many applications, and it is often included as part of standard libraries in programming languages such as C++ and Java.

## Key Concepts and Principles

The key concepts and principles of heap sort include binary heaps, heapify operations, and the extraction of the root element.

### Binary Heaps

A binary heap is a data structure that can be represented as a complete binary tree, in which each node has at most two children, and the value of each parent node is greater than or equal to the value of its children. The root node of the heap is the maximum value in the binary tree.

There are two types of binary heaps: max heaps and min heaps. In a max heap, the value of each parent node is greater than or equal to the value of its children, and the root node is the maximum value in the binary tree. In a min heap, the value of each parent node is less than or equal to the value of its children, and the root node is the minimum value in the binary tree.

### Heapify Operations

Heapify operations are used to maintain the properties of a binary heap, such as keeping the root node as the maximum or minimum value in the binary tree. There are two types of heapify operations: heapify up and heapify down.

Heapify up is used to maintain the properties of a binary heap when a node is added to the binary tree. The node is compared to its parent node, and if the value of the parent node is less than the value of the child node, the nodes are swapped. This process is repeated until the properties of the binary heap are maintained.

Heapify down is used to maintain the properties of a binary heap when a node is removed from the binary tree. The root node is removed and replaced with the last element in the binary tree, and the element is compared to its children. If the value of the child node is greater than the value of the parent node, the nodes are swapped. This process is repeated until the properties of the binary heap are maintained.

### Extraction of the Root Element

The extraction of the root element is the process of removing the maximum or minimum value from the binary heap and inserting it into a sorted array. The root node is removed and replaced with the last element in the binary tree, and heapify down operations are performed to maintain the properties of the binary heap. The extracted element is then inserted into a sorted array, and the process is repeated until all elements have been extracted and the array is sorted.

## Pseudocode and Implementation Details

The pseudocode for heap sort is as follows:

```
function heapsort(array)
n := length(array)
// Build heap (rearrange array)
for i := n/2 - 1 down to 0
heapify(array, n, i)
// One by one extract an element from heap
for i := n-1 down to 0
// Move current root to end
swap(array[0], array[i])
// call max heapify on the reduced heap
heapify(array, i, 0)
```

The `heapify`

function is defined as follows:

```
function heapify(array, n, i)
largest := i // Initialize largest as root
l := 2*i + 1 // left = 2*i + 1
r := 2*i + 2 // right = 2*i + 2
// If left child is larger than root
if l < n and array[l] > array[largest]
largest := l
// If right child is larger than largest so far
if r < n and array[r] > array[largest]
largest := r
// If largest is not root
if largest != i
swap(array[i], array[largest])
// Recursively heapify the affected sub-tree
heapify(array, n, largest)
```

Heap sort can be implemented in any programming language that supports arrays and basic sorting algorithms. In languages such as C++ and Java, heap sort is often included as part of the standard library.

## Examples and Use Cases

Heap sort can be used to sort any type of data that can be compared using a binary comparison operator, such as integers, floats, and strings. It is particularly useful for sorting large datasets that cannot be loaded into memory all at once.

Some common use cases for heap sort include:

- Sorting large databases
- Sorting arrays for data visualization
- Sorting arrays for data analysis
- Sorting arrays for computational geometry
- Sorting arrays for machine learning algorithms

## Advantages and Disadvantages

Heap sort has several advantages and disadvantages when compared to other sorting algorithms.

### Advantages

- Heap sort is an in-place algorithm that requires no additional memory allocation for sorting.
- Heap sort has a worst-case time complexity of O(n log n), which makes it very efficient for sorting large datasets.
- Heap sort is a comparison-based sorting algorithm, which makes it useful for sorting any type of data that can be compared using a binary comparison operator.

### Disadvantages

- Heap sort has a relatively slow average-case time complexity of O(n log n), which makes it less efficient than other sorting algorithms such as quick sort and merge sort.
- Heap sort has a relatively complex implementation, and may not be suitable for beginners or programmers with limited experience with data structures and algorithms.
- Heap sort is not stable, which means that it may not preserve the relative order of elements with equal keys.

## Related Algorithms or Variations

Heap sort is closely related to other sorting algorithms that use binary heaps, such as insertion sort and selection sort.

Insertion sort is a comparison-based sorting algorithm that builds the final sorted array one element at a time. It is similar to heap sort in that it maintains a sorted subarray at the beginning of the array, but insertion sort does not use a binary heap data structure.

Selection sort is a comparison-based sorting algorithm that divides the input array into two parts: the sorted part at the left end and the unsorted part at the right end. It repeatedly selects the minimum element from the unsorted part and swaps it with the leftmost unsorted element. Selection sort is similar to heap sort in that it repeatedly selects the minimum element, but selection sort does not use a binary heap data structure.