Heap Sort

Code

C language code:

#include <stdio.h>

void swap(int *a, int *b) {
    int temp = *a;
    *a = *b;
    *b = temp;
}

void heapify(int arr[], int n, int i) {
    int largest = i;
    int left = 2 * i + 1;
    int right = 2 * i + 2;

    if (left < n && arr[left] > arr[largest])
        largest = left;

    if (right < n && arr[right] > arr[largest])
        largest = right;

    if (largest != i) {
        swap(&arr[i], &arr[largest]);
        heapify(arr, n, largest);
    }
}

void heap_sort(int arr[], int n) {
    for (int i = n / 2 - 1; i >= 0; i--)
        heapify(arr, n, i);

    for (int i = n - 1; i > 0; i--) {
        swap(&arr[0], &arr[i]);
        heapify(arr, i, 0);
    }
}

int main() {
    int arr[] = {12, 11, 13, 5, 6, 7};
    int n = sizeof(arr) / sizeof(arr[0]);

    printf("Given array is \n");
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("\n");

    heap_sort(arr, n);

    printf("\nSorted array is \n");
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("\n");
    return 0;
}

Code Explanation

1. Initialization: The array arr is the array to be sorted, and n is the length of the array.
2. Swap function:
   • The swap function is used to swap two integer values.
3. Heapify function:
   • The heapify function is used to adjust a subtree into a max-heap.
   • largest is initially set to the root node i.
   • left is the left child, and right is the right child.
   • Compares the root, left child, and right child to find the maximum value.
   • If the maximum is not the root, swaps and recursively heapifies the affected subtree.
4. Heap sort function:
   • heap_sort first builds a max-heap.
   • Starting from the last non-leaf node, heapifies each node upward.
   • Then one by one moves the maximum element (root) to the end of the array and re-heapifies the remaining elements.

Example Run

Assuming the input array is {12, 11, 13, 5, 6, 7}:

1. Initial state: {12, 11, 13, 5, 6, 7}
2. Build max-heap:
   • Heapify from the last non-leaf node (index 2): {12, 11, 13, 5, 6, 7} -> {12, 11, 13, 5, 6, 7}
   • Heapify index 1: {12, 11, 13, 5, 6, 7} -> {12, 13, 11, 5, 6, 7}
   • Heapify index 0: {12, 13, 11, 5, 6, 7} -> {13, 12, 11, 5, 6, 7}
3. Start sorting:
   • Swap root with last element: {13, 12, 11, 5, 6, 7} -> {7, 12, 11, 5, 6, 13}
   • Heapify root: {7, 12, 11, 5, 6, 13} -> {12, 7, 11, 5, 6, 13}
   • Swap root with second-to-last element: {12, 7, 11, 5, 6, 13} -> {6, 7, 11, 5, 12, 13}
   • Heapify root: {6, 7, 11, 5, 12, 13} -> {11, 7, 6, 5, 12, 13}
   • Continue the above process until the array is fully sorted.
4. Final result: {5, 6, 7, 11, 12, 13}

Time Complexity Analysis

The time complexity of heap sort mainly depends on the heap-building and sorting processes.

• Best-case time complexity: $O(n \log n)$, because each heapify operation takes logarithmic time.
• Worst-case time complexity: $O(n \log n)$, regardless of the initial order of the input array, the number of steps executed is essentially the same.
• Average time complexity: $O(n \log n)$, for most input arrays, heap sort's performance is very stable.

Space Complexity Analysis

Heap sort is an in-place sorting algorithm that requires no additional auxiliary space, so the space complexity is $O(1)$.

Advantages and Disadvantages

Advantages:

• Stable time complexity; suitable for large-scale data sorting.
• No additional auxiliary space needed; low space complexity.
• Time complexity remains consistent regardless of the initial order of the input array.

Disadvantages:

• Unstable sorting algorithm; may change the relative order of equal elements.
• Relatively high implementation complexity; more complex to understand and code than simple sorting algorithms.

Applicable Scenarios

Heap sort is suitable for the following scenarios:

• Sorting large datasets.
• When a highly efficient sorting algorithm with low space complexity is needed.
• Scenarios where stable sorting is not required.

Summary

Heap sort is an efficient sorting algorithm that works by building a max-heap, moving the maximum element one by one to the end of the array, and re-heapifying the remaining elements. Although its implementation complexity is relatively high and it is unstable, it performs excellently when handling large-scale data and is a sorting algorithm with stable time complexity and low space complexity.