(C/C++)给定一个带权无序数组，线性复杂度求出其带权中位数(select分治算法)

给定一个未排序的数组(x1, x2, … ,xn)，其中每个元素关联一个权值：(w1, w2, … ,wn)，且 。请设计一个线性时间的算法，在该数组中查找其带权中位数xk，满足：

image

思路

基于寻找无序数组第k小个数的select算法，以rand()选出的pivot将数组分为三个部分，并进行前后两部分权值总和的判断。
若leftWeight <=0.5 && rightWeight <=0.5，pivot为带权中位数
否则，若leftWeight > rightWeight，则带权中位数在pivot左侧数组中，将右侧权值赋给pivot，进行递归
leftWeight<= rightWeight同理

伪代码

3.png

#include 
#include 
#include 
#include 

#define N 5
using namespace std;

struct node
{
    int value;
    double weight;
};

const int VALUE_MAX = 100;
node INDEX[N] = { {1,0.6},{2,0.1},{5,0.1},{3,0.1},{4,0.1} };

void Print(node *A, int len)
{
    int i;
    for (i = 0; i < len; i++)
        cout << A[i].value << '\t';
    cout << endl;
    for (i = 0; i < len; i++)
        cout << A[i].weight << '\t';
    cout << endl;
}

//返回选定的pivot中值
int Partition(node *A, int begin, int end)
{
    int less = begin - 1, i;
    int pivotIndex = begin + rand() % (end - begin + 1);
    for (i = begin; i <= end; i++)
    {
        if (A[i].value < A[pivotIndex].value)
        {
            less++;
            swap(A[less], A[i]);
        }
    }
    swap(A[less + 1], A[pivotIndex]);
    return less + 1;
}

double getSumWeight(node*A, int begin, int end) {
    double sum = 0;
    for (int i = begin; i <= end; i++) {
        sum += A[i].weight;
    }
    return sum;
}

//
int selectWeightedMedian(node* index, int begin, int end) {
    if (begin == end)
        return index[begin].value;
    if (end - begin == 1) {
        if (index[begin].weight == index[end].weight)
            return (index[begin].value + index[end].value) / 2;
        if (index[begin].weight > index[end].weight)
            return index[begin].value;
        else
            return index[end].value;
    }
    int midIndex = Partition(index, begin, end);
    double leftWeight = getSumWeight(index, begin, midIndex - 1);
    double rightWeight = getSumWeight(index, midIndex + 1, end);
    if (leftWeight <= 0.5&&rightWeight <= 0.5)
        return index[midIndex].value;
    else
        if (leftWeight > rightWeight) {
            index[midIndex].weight += rightWeight;
            return selectWeightedMedian(index, begin, midIndex);
        }
        else {
            index[midIndex].weight += leftWeight;
            return selectWeightedMedian(index, midIndex, end);
        }
}

int main(void) {
    srand((int)time(0));
    cout << setprecision(3);
    int length, sum = 0;
    cout << "请输入数组长度:";
    cin >> length;
    node *index = new node[length + 1];
    int * weights = new int[length + 1];
    //生成随机数据
    for (int i = 0; i < length; i++)
    {
        index[i].value = rand() % VALUE_MAX;
        do { weights[i] = rand() % VALUE_MAX; } while (weights[i] == 0);
        sum = sum + weights[i];
    }
    //将权值规格化
    for (int i = 0; i < length; i++)
        index[i].weight = (double)weights[i] / sum;
    //打印生成的数据
    Print(index, length);
    cout << "带权中位数为:" << selectWeightedMedian(index, 0, length - 1) << endl;
    system("pause");
    return 0;
}

运行示例

运行示例