(C/C++)給定一個帶權無序陣列，線性複雜度求出其帶權中位數(select分治演算法)

摘要： 給定一個未排序的陣列(x1, x2, … ,xn)，其中每個元素關聯一個權值：(w1, w2, … ,wn)，且 。請設計一個線性時間的演算法，在該陣列中查詢其帶權中位數xk，滿足： image 思路 基於尋找無序陣...

思路

基於尋找無序陣列第k小個數的select演算法，以rand()選出的pivot將陣列分為三個部分，並進行前後兩部分權值總和的判斷。

若leftWeight <=0.5 && rightWeight <=0.5，pivot為帶權中位數

否則，若leftWeight > rightWeight，則帶權中位數在pivot左側陣列中，將右側權值賦給pivot，進行遞迴

leftWeight<= rightWeight同理

虛擬碼

#include <iostream>
#include <iomanip>
#include <stdlib.h>
#include <time.h>

#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;
}

//返回選定的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 << "請輸入陣列長度:";
cin >> length;
node *index = new node[length + 1];
int * weights = new int[length + 1];
//生成隨機資料
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];
}
//將權值規格化
for (int i = 0; i < length; i++)
index[i].weight = (double)weights[i] / sum;
//列印生成的資料
Print(index, length);
cout << "帶權中位數為:" << selectWeightedMedian(index, 0, length - 1) << endl;
system("pause");
return 0;
}

執行示例