堆排序是一种基于二叉堆数据结构的比较排序技术。它类似于选择排序，我们首先找到最小元素并将最小元素放在开头。我们对其余元素重复相同的过程。
二叉堆是什么？
让我们首先定义一个完整的二叉树。完全二叉树是一棵二叉树，其中每一层（可能除了最后一层）都被完全填充，并且所有节点都尽可能地向左（来源维基百科）
二叉堆是一棵完全二叉树，其中项目以特殊顺序存储，使得父节点中的值大于（或小于）其两个子节点中的值。前者称为最大堆，后者称为最小堆。堆可以用二叉树或数组表示。
为什么二进制堆的基于数组的表示？
由于二叉堆是一棵完整的二叉树，它可以很容易地表示为一个数组，并且基于数组的表示是节省空间的。如果父节点存储在索引 I 处，则可以通过 2 * I + 1 计算左子节点，通过 2 * I + 2 计算右子节点（假设索引从 0 开始）。
堆排序算法按升序排序：
1.从输入数据构建最大堆。
2.此时，最大的项存储在堆的根部。将其替换为堆的最后一项，然后将堆的大小减 1。最后，堆化树的根。
3.当堆的大小大于 1 时，重复步骤 2。
如何构建堆？
只有当其子节点被堆化时，才能将 Heapify 过程应用于节点。所以堆化必须按照自下而上的顺序进行。
让我们通过一个例子来理解：
Input data: 4, 10, 3, 5, 1
4(0)
/ \
10(1) 3(2)
/ \
5(3) 1(4)
The numbers in bracket represent the indices in the array
representation of data.
Applying heapify procedure to index 1:
4(0)
/ \
10(1) 3(2)
/ \
5(3) 1(4)
Applying heapify procedure to index 0:
10(0)
/ \
5(1) 3(2)
/ \
4(3) 1(4)
The heapify procedure calls itself recursively to build heap
in top down manner.C++
// C++ program for implementation of Heap Sort
#include
using namespace std;
// To heapify a subtree rooted with node i which is
// an index in arr[]. n is size of heap
void heapify(int arr[], int n, int i)
{
int largest = i; // Initialize largest as root
int l = 2 * i + 1; // left = 2*i + 1
int r = 2 * i + 2; // right = 2*i + 2
// If left child is larger than root
if (l < n && arr[l] > arr[largest])
largest = l;
// If right child is larger than largest so far
if (r < n && arr[r] > arr[largest])
largest = r;
// If largest is not root
if (largest != i) {
swap(arr[i], arr[largest]);
// Recursively heapify the affected sub-tree
heapify(arr, n, largest);
}
}
// main function to do heap sort
void heapSort(int arr[], int n)
{
// Build heap (rearrange array)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
// One by one extract an element from heap
for (int i = n - 1; i > 0; i--) {
// Move current root to end
swap(arr[0], arr[i]);
// call max heapify on the reduced heap
heapify(arr, i, 0);
}
}
/* A utility function to print array of size n */
void printArray(int arr[], int n)
{
for (int i = 0; i < n; ++i)
cout << arr[i] << " ";
cout << "\n";
}
// Driver code
int main()
{
int arr[] = { 12, 11, 13, 5, 6, 7 };
int n = sizeof(arr) / sizeof(arr[0]);
heapSort(arr, n);
cout << "Sorted array is \n";
printArray(arr, n);
} Java
// Java program for implementation of Heap Sort
public class HeapSort {
public void sort(int arr[])
{
int n = arr.length;
// Build heap (rearrange array)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
// One by one extract an element from heap
for (int i = n - 1; i > 0; i--) {
// Move current root to end
int temp = arr[0];
arr[0] = arr[i];
arr[i] = temp;
// call max heapify on the reduced heap
heapify(arr, i, 0);
}
}
// To heapify a subtree rooted with node i which is
// an index in arr[]. n is size of heap
void heapify(int arr[], int n, int i)
{
int largest = i; // Initialize largest as root
int l = 2 * i + 1; // left = 2*i + 1
int r = 2 * i + 2; // right = 2*i + 2
// If left child is larger than root
if (l < n && arr[l] > arr[largest])
largest = l;
// If right child is larger than largest so far
if (r < n && arr[r] > arr[largest])
largest = r;
// If largest is not root
if (largest != i) {
int swap = arr[i];
arr[i] = arr[largest];
arr[largest] = swap;
// Recursively heapify the affected sub-tree
heapify(arr, n, largest);
}
}
/* A utility function to print array of size n */
static void printArray(int arr[])
{
int n = arr.length;
for (int i = 0; i < n; ++i)
System.out.print(arr[i] + " ");
System.out.println();
}
// Driver code
public static void main(String args[])
{
int arr[] = { 12, 11, 13, 5, 6, 7 };
int n = arr.length;
HeapSort ob = new HeapSort();
ob.sort(arr);
System.out.println("Sorted array is");
printArray(arr);
}
}Python
# Python program for implementation of heap Sort
# To heapify subtree rooted at index i.
# n is size of heap
def heapify(arr, n, i):
largest = i # Initialize largest as root
l = 2 * i + 1 # left = 2*i + 1
r = 2 * i + 2 # right = 2*i + 2
# See if left child of root exists and is
# greater than root
if l < n and arr[largest] < arr[l]:
largest = l
# See if right child of root exists and is
# greater than root
if r < n and arr[largest] < arr[r]:
largest = r
# Change root, if needed
if largest != i:
arr[i], arr[largest] = arr[largest], arr[i] # swap
# Heapify the root.
heapify(arr, n, largest)
# The main function to sort an array of given size
def heapSort(arr):
n = len(arr)
# Build a maxheap.
for i in range(n//2 - 1, -1, -1):
heapify(arr, n, i)
# One by one extract elements
for i in range(n-1, 0, -1):
arr[i], arr[0] = arr[0], arr[i] # swap
heapify(arr, i, 0)
# Driver code
arr = [12, 11, 13, 5, 6, 7]
heapSort(arr)
n = len(arr)
print("Sorted array is")
for i in range(n):
print("%d" % arr[i]),
# This code is contributed by Mohit KumraC#
// C# program for implementation of Heap Sort
using System;
public class HeapSort {
public void sort(int[] arr)
{
int n = arr.Length;
// Build heap (rearrange array)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
// One by one extract an element from heap
for (int i = n - 1; i > 0; i--) {
// Move current root to end
int temp = arr[0];
arr[0] = arr[i];
arr[i] = temp;
// call max heapify on the reduced heap
heapify(arr, i, 0);
}
}
// To heapify a subtree rooted with node i which is
// an index in arr[]. n is size of heap
void heapify(int[] arr, int n, int i)
{
int largest = i; // Initialize largest as root
int l = 2 * i + 1; // left = 2*i + 1
int r = 2 * i + 2; // right = 2*i + 2
// If left child is larger than root
if (l < n && arr[l] > arr[largest])
largest = l;
// If right child is larger than largest so far
if (r < n && arr[r] > arr[largest])
largest = r;
// If largest is not root
if (largest != i) {
int swap = arr[i];
arr[i] = arr[largest];
arr[largest] = swap;
// Recursively heapify the affected sub-tree
heapify(arr, n, largest);
}
}
/* A utility function to print array of size n */
static void printArray(int[] arr)
{
int n = arr.Length;
for (int i = 0; i < n; ++i)
Console.Write(arr[i] + " ");
Console.Read();
}
// Driver code
public static void Main()
{
int[] arr = { 12, 11, 13, 5, 6, 7 };
int n = arr.Length;
HeapSort ob = new HeapSort();
ob.sort(arr);
Console.WriteLine("Sorted array is");
printArray(arr);
}
}
// This code is contributed
// by Akanksha Rai(Abby_akku)PHP
$arr[$largest])
$largest = $l;
// If right child is larger than largest so far
if ($r < $n && $arr[$r] > $arr[$largest])
$largest = $r;
// If largest is not root
if ($largest != $i)
{
$swap = $arr[$i];
$arr[$i] = $arr[$largest];
$arr[$largest] = $swap;
// Recursively heapify the affected sub-tree
heapify($arr, $n, $largest);
}
}
// main function to do heap sort
function heapSort(&$arr, $n)
{
// Build heap (rearrange array)
for ($i = $n / 2 - 1; $i >= 0; $i--)
heapify($arr, $n, $i);
// One by one extract an element from heap
for ($i = $n-1; $i > 0; $i--)
{
// Move current root to end
$temp = $arr[0];
$arr[0] = $arr[$i];
$arr[$i] = $temp;
// call max heapify on the reduced heap
heapify($arr, $i, 0);
}
}
/* A utility function to print array of size n */
function printArray(&$arr, $n)
{
for ($i = 0; $i < $n; ++$i)
echo ($arr[$i]." ") ;
}
// Driver program
$arr = array(12, 11, 13, 5, 6, 7);
$n = sizeof($arr)/sizeof($arr[0]);
heapSort($arr, $n);
echo 'Sorted array is ' . "\n";
printArray($arr , $n);
// This code is contributed by Shivi_Aggarwal
?>Javascript
输出
Sorted array is
5 6 7 11 12 13 这是以前的 C 代码供参考。
笔记：
堆排序是一种就地算法。
它的典型实现是不稳定的，但可以变得稳定（见这个）
时间复杂度： heapify 的时间复杂度为 O(Logn)。 createAndBuildHeap() 的时间复杂度为 O(n)，Heap Sort 的整体时间复杂度为 O(nLogn)。
堆排序的应用
1.对一个近似排序（或K排序）的数组进行排序
2.数组中的 k 个最大（或最小）元素
堆排序算法的用途有限，因为 Quicksort 和 Mergesort 在实践中更好。然而，堆数据结构本身被大量使用。查看堆数据结构的应用
https://youtu.be/MtQL_ll5KhQ
快照：
堆排序测验
GeeksforGeeks/GeeksQuiz 上的其他排序算法：
快速排序、选择排序、冒泡排序、插入排序、归并排序、堆排序、快速排序、基数排序、计数排序、桶排序、壳排序、梳状排序、鸽洞排序
排序的编码实践。
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