算法导论6.57习题解答

CLRS 6.5-7 ：
HEAP-DELETE(A, i)操作将结点i中的项从堆A中删去。对含n个元素的最大堆，请给出时间为O(lgn)的HEAP-DELETE的实现。
算法思想：
对即将被删的i处的元素设置为堆中末尾的元素，同时堆的大小减1.
此题同时附带6.2-5的解答
代码如下：

#include <iostream>
using namespace std;
//修正i的位置，在此处已经假设i的子节点都是堆
static void max_heapify(int*&a, int i, int length);
//建立数组的堆
void build_max_heap(int*&a, int length);
//利用堆对数组重新排序，总是拿第一个和最后一个对调，数组长度减一
void heap_sort(int*&a, int length);
//删除ith元素
void heap_delete(int*&a, int i, int length);
//插入x
void heap_insert(int*&a, int x, int length);
//将ith位置上的数增加到key
void increase_key(int*&a, int i, int key);

int main()
{
    const int LEN =10;
    int b[10] = {12, 43, 0, -4, 98, 75, 64, 88, 5, 32};
    int* a = new int[LEN];
    for(int i =0; i < LEN; i++)
        a[i] = b[i];
    heap_sort(a, LEN);
    for(int i =0; i < LEN; i++)
        cout<<a[i]<<endl;
    cout<<"****"<<endl;

    increase_key(a, 7, 100);
    heap_sort(a, LEN);
    for(int i =0; i < LEN; i++)
        cout<<a[i]<<endl;
    cout<<"****"<<endl;

    heap_insert(a, 10, LEN);
    heap_sort(a, LEN + 1);
    for(int i =0; i < LEN + 1; i++)
        cout<<a[i]<<endl;
    cout<<"****"<<endl;

    heap_delete(a, 1, LEN + 1);
    heap_sort(a, LEN);
    for(int i =0; i < LEN; i++)
        cout<<a[i]<<endl;
    cout<<"****"<<endl;
    return 0;
}

static void max_heapify(int*&a, int i, int length)
{
    int largest = i;
    while(largest <= length -1) {
        int left =2*largest +1;
        int right =2*largest +2;
        int temp = largest;
        if(left <= length -1&& a[left] > a[largest]) {
            largest = left;
        }
        if(right <= length -1&& a[right] > a[largest]) {
            largest = right;
        }
        if(largest != temp) {
            int exchange = a[largest];
            a[largest] = a[temp];
            a[temp] = exchange;
        } else {
            break;
        }
    }
}

void build_max_heap(int*&a, int length)
{
    int root = length/2-1;
    for(int i = root; i >=0; i--)
        max_heapify(a, i, length);
}

void heap_sort(int*&a, int length)
{
    build_max_heap(a, length);
    for(int i = length -1; i >=1; i--) {
        int temp = a[0];
        a[0] = a[i];
        a[i] = temp;
        max_heapify(a, 0, i);
    }
}

void heap_delete(int*&a, int i, int length)
{
    if(i != length -1) {
        a[i] = a[length -1];
        max_heapify(a, i, length); 
    }
}

void heap_insert(int*&a, int x, int length)
{
    int* temp = a;
    a = new int[length +1];
    for(int i =0; i < length; i++)
        a[i] = temp[i];
    delete temp;
    //用-10000000代替负无穷
    a[length] =-10000000;
    increase_key(a, length, x);
}

void increase_key(int*&a, int i, int key)
{
    if(a[i] > key) {
        cout<<"key should be larger than a[i]"<<endl;
    } else {
        int parent = (i -1)/2;
        a[i] = key;
        while(parent >= 0 && a[parent] < key) {
            int temp = a[parent];
            a[parent] = key;
            a[i] = temp;
            i = parent;
            parent = (i -1)/2;
        }
    }
}