堆排序 (Heap Sort)

1、堆的性质：堆是一种数组，但是以树的结构形式来看待它，如下标 i 节点的求解Parent和Children节点如下：

PARENT(i)
return ⌊i/2⌋
LEFT(i)
return 2i
RIGHT(i)
return 2i + 1

最大堆：根以外的每个结点i都不大于其根结点，即根为最大元素，在顶端，有A[PARENT(i)] （根）≥ A[i]

最小堆：根以外的每个结点i都不小于其根结点，即根为最小元素，在顶端，有A[PARENT(i)] （根）≤ A[i]

堆排序时间复杂度：O(nlogn)

2、堆的维持 (nlogn)：

MAX-HEAPIFY(A, i)1 l ← LEFT(i)2 r ← RIGHT(i)3 if l ≤ heap-size[A] and A[l] > A[i]4    then largest ← l5    else largest ← i6 if r ≤ heap-size[A] and A[r] > A[largest]7    then largest ← r8 if largest ≠ i9    then exchange A[i] ↔ A[largest]10         MAX-HEAPIFY(A, largest)

3、建堆:不断调用MAX-HEAPIFY函数来建堆

BUILD-MAX-HEAP(A)1  heap-size[A] ← length[A]2  for i ← ⌊length[A]/2⌋ downto 13       do MAX-HEAPIFY(A, i)

4、堆排序：首先是把一个给定的数组变成MAX堆，然后把根节点和堆的最后一个交换，堆的大小减1，再调整堆，这样下去，直到堆的大小为1：伪代码如下：[/code]

HEAPSORT(A)1 BUILD-MAX-HEAP(A)2 for i ← length[A] downto 23    do exchange A[1] ↔ A[i]4       heap-size[A] ← heap-size[A] - 15       MAX-HEAPIFY(A, 1)

堆排序的图解过程如下：

5、Java源代码：

package Sort;public class HeapSort {//Private variable to record the size of heapprivate static int heapSize;//Heap sort methodpublic static float[] HeapSortAlgorithm(float flt[], int length){Build_Max_Heap(flt, length);for(int i=length-1;i>0;i--){Swap(flt, 0, i);heapSize--;Max_Heapify(flt, 0);}return flt;}//Build the max heap from input arraypublic static void Build_Max_Heap(float flt[], int length){heapSize=length;for (int i=(length-1)/2;i>=0;i--) {Max_Heapify(flt, i);}}//Adjust the position of elements in the array to satisfy the max heapprivate static void Max_Heapify(float flt[], int i){int leftChild=2*i+1;int rightChild=2*i+2;int largest=i;if (leftChild<heapSize&&flt[leftChild]>flt[i]) {largest=leftChild;}if (rightChild<heapSize&&flt[rightChild]>flt[largest]) {largest=rightChild;}if (largest!=i) {Swap(flt, largest, i);Max_Heapify(flt, largest);}}//Swap the values of two elements in the arrayprivate static void Swap(float flt[], int left, int right){float temp=flt[left];flt[left]=flt[right];flt[right]=temp;}}

另一个版本Java Code：

public class HeapTest {/*** @param args*/public static void buildHeap(int arr[],int startIndex, int range){int value = arr[startIndex];int large = startIndex*2+1;while(large <= range){if(large < range && arr[large]>arr[large+1]){large++;}if(arr[large]<value){arr[startIndex] = arr[large];arr[large] = value;startIndex = large;large = startIndex*2+1;}else {break;}}}public static void HeapSort(int arr[],int start, int end){for(int i=end/2;i>=0;i--){buildHeap(arr, i, end-1);}int endTemp = end;for(int i=0;i<endTemp;i++){System.out.println(arr[0]);arr[0] = arr[end - 1];buildHeap(arr, 0, end -1);end--;}}public static void main(String[] args) {// TODO Auto-generated method stubint nums[] = {9,8,7,6,5,4,3,2,1,0};HeapSort(nums, 0, 10);}}

参考：

http://blog.csdn.net/v_JULY_v/article/details/6198644[/code]

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