算法导论（4） 顺序统计量

1.最大值与最小值
将一对输入元素相互进行比较，然后把较小的与当前的最小值比较，把较大的与当前最大值比较，这样每两个元素比较3次，最多需要3*n/2次比较可以同时找到一组输入元素中的最大值与最小值。
2.顺序统计量
一个n个元素组成的集合中，第i个顺序统计量是该集合中第i小的元素。中位数是i=(n+1)/2处的元素。
期望为线性时间的选择算法RandomSelect与最坏情况为线性时间的选择算法select如下：

#include<iostream>
using namespace std;
#include<CTime>
#include<vector>
template<class T>
void QuickSort2(T a[], int left, int right)
{
    T temp, x = a[right];
    int i, j, q;
    if (left < right){
        i = left - 1;
        for (j = left; j < right; j++)
        {
            if (a[j] <= x)
            {
                i += 1;
                temp = a[j];
                a[j] = a[i];
                a[i] = temp;
            }
        }
        temp = a[right];
        a[right] = a[i + 1];
        a[i + 1] = temp;
        q = i + 1;
        QuickSort2(a, left, q - 1);
        QuickSort2(a, q + 1, right);
    }
}
int RandomPartition(int a[], int left, int right)
{
    srand(time(NULL));
    int m = left + rand() % (right - left);
    int temp =a[right];
    a[right] = a[m];
    a[m] = temp;
    int x = a[right];
    int i, j;
    i = left - 1;
    for (j = left; j < right; j++)
    {
        if (a[j] <= x)
        {
            i += 1;
            temp = a[j];
            a[j] = a[i];
            a[i] = temp;
        }
    }
    temp = a[right];
    a[right] = a[i + 1];
    a[i + 1] = temp;
    return i + 1;

}

int RandomSelect(int a[], int left, int right, int i)
{
    if (left == right){ return a[left]; }
    int q = RandomPartition(a, left, right);
    int k = q - left + 1;
    if (i == k)
    {
        return a[q];
    }
    else if (i < k){
        return RandomSelect(a, left, q - 1, i);
    }
    else{
        return RandomSelect(a, q + 1, right, i - k);
    }
}

int Partition(vector<int>&a, int left, int right,int b)
{
    int m = 0;
    for (int i = left; i <= right; i++){
        if (a[i] == b){
            m = i;
            break;
        }
    }

    int temp = a[right];
    a[right] = a[m];
    a[m] = temp;
    int x = a[right];
    int i, j;
    i = left - 1;
    for (j = left; j < right; j++)
    {
        if (a[j] <= x)
        {
            i += 1;
            temp = a[j];
            a[j] = a[i];
            a[i] = temp;
        }
    }
    temp = a[right];
    a[right] = a[i + 1];
    a[i + 1] = temp;
    return i + 1;

}

//原数组使用vector是因为寻找中位数的中位数时的数组不固定，但也需要调用select函数
int select(vector<int>&a, int left, int right, int dst)
{
    int n = right - left + 1;
    if (n == 1){ return a[left]; }
    vector<int> temp;
    vector<int> b;

    for (int i = left; i <right+1; i += 5)
    {
        if (right+1 - i < 5){ break; }
        for (int j = i; j - i < 5; j++)
        {
            b.push_back(a[j]);//不能再原址进行排序，否则可能会改变分割后的数组
        }
        //插入排序
        for (int j = 1; j < 5; j++)
        {
            int t = b[j];
            int m = j - 1;
            while (m >= 0 && b[m]>t)
            {
                b[m + 1] = b[m];
                m--;
            }
            b[m + 1] = t;
        }

        temp.push_back(b[2]);
        b.clear();
    }
    if (n%5!= 0){
        for (int i =left+ (n / 5) * 5 ; i < right+1; i++)
        {
            b.push_back(a[i]);
        }
        //插入排序
        for (int i = 1; i<b.size();i++)
        {
            int t = b[i];
            int j = i - 1;
            while (j >=0 && b[j]>t)
            {
                b[j + 1] = b[j];
                j--;
            }
            b[j + 1] = t;
        }
        temp.push_back(b[(b.size()+1)/2-1]);
        b.clear();
    }

    int q = Partition(a, left, right, select(temp, 0, temp.size()-1, (temp.size()+1) / 2));
    int k = q - left + 1;
    if (dst == k)
    {
        return a[q];
    }
    else if (dst < k){
        return select(a, left, q - 1, dst);
    }
    else{
        return select(a, q + 1, right, dst - k);
    }
}
int main()
{
    int a[22];
    vector<int> m;
    srand(time(NULL));
    cout << "原数组为：" << endl;
    for (int i = 0; i < 22; i++)
    {
        a[i] = rand() % 100;
        m.push_back(a[i]);
        cout << a[i] << " ";

    }
    cout << endl;

    int num =12;
    int b=RandomSelect(a, 0, 21, num);
    cout << "方法一：第"<<num<<"小的数为："<<b << endl;
    b = select(m, 0, 21, num);
    cout << "方法二：第" << num << "小的数为：" << b << endl;
    QuickSort2<int>(a, 0, 21);
    cout << "快速排序后为：" << endl;
    for (int i = 0; i < 22; i++)
    {
        cout << a[i] << " ";
    }
    cout << endl;
}