堆排序

二叉堆是一个数组，它可以被看成一个近似的完全二叉树。树上的每个节点对应对应数组中的一个元素。堆结构上的一些基本操作的运行时间至多与树的高度成正比，即时间复杂度O(lgn)。

维护堆的性质、建堆、堆排序的代码如下：

/*堆排序
 * 二叉堆是一个数组，它可以被看成一个近似的完全二叉树
 * */

/*定义数组类型*/
typedef int arr_type;

/*定义堆结构体*/
typedef struct heap_struct
{
	arr_type *array;
	int arr_length;
	int heap_size;
} heap;

/*定义宏，分别计算父节点、左孩子、右孩子的下标*/
#define PARENT(i) (i / 2) 
#define LEFT(i) (2 * i)
#define RIGHT(i) (2 * i + 1)


/*维护最大堆的性质
 * @heap_arg 指向一个堆结构的指针
 * @i 根结点下标*/
void max_heapify(heap *heap_arg, int i)
{
	int l = LEFT(i);
	int r = RIGHT(i);
	int largest = i; /*用来记录比根结点大的元素的下标*/

	arr_type ele_l = (heap_arg->array)[l]; /*左孩子*/
	arr_type ele_r = (heap_arg->array)[r]; /*右孩子*/
	arr_type ele_i = (heap_arg->array)[i]; /*根结点*/
	arr_type tmp;

	if (l <= heap_arg->heap_size && ele_l > ele_i)
	{
		largest = l;
	}

	if (r <= heap_arg->heap_size && ele_r > (heap_arg->array)[largest])
	{
		largest = r;
	}

	if (largest != i)
	{
		tmp = (heap_arg->array)[largest];
		(heap_arg->array)[largest] = (heap_arg->array)[i];
		(heap_arg->array)[i] = tmp;
		max_heapify(heap_arg, largest);
	}

}

/*建堆
 * 注：子数组A(floor(n/2) + 1 ... n)中的元素都是树的叶节点
 * @heap_arg 指向堆结构的指针*/
void build_max_heap(heap *heap_arg)
{
	heap_arg->heap_size = heap_arg->arr_length;
	int i;
	for (i = heap_arg->arr_length / 2; i > 0; i--)
	{
		max_heapify(heap_arg, i);
	}

}

/*堆排序算法
 * @heap_arg 待排序的堆
 */
void max_heap_sort(heap *heap_arg)
{
	heap_arg->heap_size = heap_arg->arr_length;
	arr_type tmp;

	build_max_heap(heap_arg); /*构建最大堆*/

	int i;
	for (i = heap_arg->arr_length; i > 1; i--)
	{
		tmp = (heap_arg->array)[i];
		(heap_arg->array)[i] = (heap_arg->array)[1];
		(heap_arg->array)[1] = tmp;
		heap_arg->heap_size = heap_arg->heap_size - 1;
		max_heapify(heap_arg, 1);
	}
	
}

测试如下：

/*测试堆排序*/
#include <stdio.h>
#include <stdlib.h>
#include "heap_sort.h"

int main(void)
{
	arr_type arr_a[13] = {0, 15, 13, 9, 5, 12, 8, 7, 4, 0, 6, 2, 1};	

	heap *heap_a = (heap *) malloc(sizeof(heap));
	heap_a->array = arr_a;
	heap_a->arr_length = 12;
	heap_a->heap_size = heap_a->arr_length;

	printf("The array is ");
	for (int i = 1; i < 13; i++)
	{
		printf("%d ", arr_a[i]);
	}
	printf("
");

	build_max_heap(heap_a);
	printf("The max heap is ");
	for (int i = 1; i < 13; i++)
	{
		printf("%d ", arr_a[i]);
	}
	printf("
");

	printf("Test max_heapify(): 
");
	for (int i = 12; i > 1; i--)
	{
		int tmp = (heap_a->array)[i];
		(heap_a->array)[i] = (heap_a->array)[1];
		(heap_a->array)[1] = tmp;
		for (int j = 1; j < 13; j++)
		{
			printf("%d ", arr_a[j]);
		}
		heap_a->heap_size -= 1;
		max_heapify(heap_a, 1);
		printf("
");
	}	

	printf("Test max_heap_sort(): 
");
	max_heap_sort(heap_a);
	for (int i = 1; i < 13; i++)
	{
		printf("%d ", arr_a[i]);
	}
	printf("
");


	free(heap_a);

	return 1;
}