算法导论 第六章：优先级队列

        虽然堆排序的时间复杂度为O(nlgn),但在实际中，快速排序(下一章将介绍)往往要优于堆排序。尽管如此，堆数据结构在其他方面有着很大的用处，如用于优先级队列的实现。因为堆可让优先级队列的所有操作的时间复杂度为O(lgn)。最大优先级队列常用于共享主机上的作业调度，最小优先级队列常用于事件驱动的模拟中。

以最大优先级队列为例，其支持的操作如下:

1)INSERT(S,x): 向集合中插入元素x



2)MAXIMUM(S): 返回集合S的最大值



3)EXTRACT-MAX(S):删除集合S的最大值



4)INCREASE-KEY(S,x,k):将集合S中的元素x的值增加为k



最大优先级队列实现的完整代码如下：

#include<iostream>
#include<cstdlib>
#include<climits>
#define PARENT(i) (i/2)
#define LEFT(i)   (2*i)
#define RIGHT(i)  (2*i+1)
using namespace std;

void Print(int *a)
{
int n=a[0];
for(int i=1;i<=n;i++)
cout<<a[i]<<"   ";
cout<<endl;
}
int *Transform(int *a,int n)
{
int * A=new int[n+1];
A[0]=n;
for(int i=0;i<n;i++)
A[i+1]=a[i];
return A;
}
void swap(int &a,int &b)
{
int temp;
temp=a;
a=b;
b=temp;
}
void Max_Heapify(int *A,int i)
{//adjust the heap to maintain the heap property
int heap_size=A[0];
int l=LEFT(i) ;   //index of i's left child
int r=RIGHT(i);
int largest;     //At each step,the largest of the element A[i],A[LEFT(i)]
//A[RIGHT(i)] is determined,and its index is stored in largest.
if(l<=heap_size && A[l]>A[i])
largest	= l;
else
largest = i;

if(r<=heap_size && A[r]>A[largest])
largest = r;

if(largest!=i)     //the subtree rooted at i violate the max-heap property
{
swap(A[i],A[largest]);
Max_Heapify(A,largest);  //after exchanging ,the subtree rooted at largest might violate the max-heap property
}
}
void Build_MaxHeap(int *A)
{//Transform array A into max-heap A,where A[0]=heap-size
int A_length=A[0];
for(int i=A_length/2;i>=1;i--)
Max_Heapify(A,i);
}
int GetHeapMaximum(int *A)
{//return the elements of S with the largest key
Build_MaxHeap(A);
return A[1];
}
int Heap_ExtractMax(int *A)
{
int max;
int heap_size;
heap_size=A[0];
if(heap_size<1)
{
cout<<"heap underflow"<<endl;
return -1;
}
Build_MaxHeap(A);
max=A[1];
A[1]=A[heap_size];
heap_size--;
A[0]=heap_size;
Max_Heapify(A,1);

return max;
}
void Heap_IncreaseKey(int *A,int i,int key)
{
Build_MaxHeap(A);
if(key<A[i]){
cout<<"new key is smaller than the current key."<<endl;
return;
}
A[i]=key;
while(i>1 && A[PARENT(i)]<A[i])
{
swap(A[PARENT(i)],A[i]);
i=PARENT(i);
}
}
void Heap_InsertKey(int *A,int key)
{
int heap_size=A[0];
heap_size=heap_size+1;
A[0]=heap_size;
A[heap_size]=INT_MIN;
Heap_IncreaseKey(A,heap_size,key);
}
int main(){

int a[]={4,1,3,2,16,9,10,14,8,7};
int n=sizeof(a)/sizeof(int);
int *A=new int[n+1];

cout<<"---------------Init the set--------------------"<<endl;
cout<<"After initing,S[1..n] is:"<<endl;
A=Transform(a,n);   //Transform a[0...n-1] into a[1...n];a[0]=a.length
Print(A);

cout<<"---------------Excute MAXIMUM------------------"<<endl;
int hMaxN;
hMaxN=GetHeapMaximum(A);
cout<<"The largest key is:"<<hMaxN<<endl;

cout<<"------------Excute EXTRACT-MAX-----------------"<<endl;
hMaxN=Heap_ExtractMax(A);
cout<<"The largest key is:"<<hMaxN<<endl;
cout<<"After removing the largest,the heap is:"<<endl;
Print(A);

cout<<"------------Excute INCREASE-KEY----------------"<<endl;
int key=11;
int index=6;
Heap_IncreaseKey(A,index,key);
cout<<"After increasing ,the heap is:"<<endl;
Print(A);

cout<<"--------------Excute HEAP-INSERT---------------"<<endl;
int ikey=13;
Heap_InsertKey(A,ikey);
Print(A);
cout<<"-----------------------------------------------"<<endl;
return 0;
}

运行结果如下：



若有错误，请指正~~~