Program to find weighted median of a given array

Given two arrays arr[] of N integers and W[] of N weights where W[i] is the weight for the element arr[i]. The task is to find the weighted median of the given array.
Note: The sum of the weight of all elements will always be 1.

Let the array arr[] be arranged in increasing order with their corresponding weights.

If N is odd, then there is only one weighted median say arr[k] which satisfies the below property:

\sum _{i=1}^{k-1}W_{i}\leq 1/2 \;and\; \sum _{i=k+1}^{N}W_{i}\leq 1/2

If N is even, then there are two weighted medians, i.e., lower and upper weighted median.



The lower weighted median for element arr[k] which satisfies the following:

\sum _{i=1}^{k-1}W_{i}< 1/2 \;and\; \sum _{i=k+1}^{N}W_{i}= 1/2

The upper weighted median for element arr[k] which satisfies the following:

\sum _{i=1}^{k-1}W_{i}= 1/2 \;and\; \sum _{i=k+1}^{N}W_{i}< 1/2

Examples:

Input: arr={5, 1, 3, 2, 4}, W=[0.25, 0.15, 0.2, 0.1, 0.3]
Output: The weighted median is element 4
Explanation:
Here the number of element is odd, so there is only one weighted median because at K = 3 the above condition is satisfied.
The cumulative weights on each side of element 4 is 0.45 and 0.25.

Input: arr=[4, 1, 3, 2], W=[0.25, 0.49, 0.25, 0.01]
Output:
The lower weighted median is element 2
The upper weighted median is element 3
Explanation: 
Here there are an even number of elements, so there are two weighted medians.
Lower weighted median is at K = 2 because at K = 2 the above condition is satisfied with cumulative weight on each side of element 2 is 0.49 and 0.5.
Upper weighted median is at K = 3 because at K = 3 the above condition is satisfied with cumulative weight on each side of element 3 is 0.5 and 0.25.

Approach: Follow the steps below to solve the given problem:

  1. Now to find the median of the array arr[] in increasing order with their respective order of weight shouldn’t be changed.
  2. So, create a set of pairs where the first element of the pair will be arr[i] and the second element of the pair will be its corresponding weights W[i].
  3. Then sort the set of Pairs according to the arr[] values.
  4. If the number of pairs is odd, then find the weighted median as:
    • Traverse over the set of pairs and compute sum by adding weights.
    • When the sum becomes greater than 0.5 print the arr[i] value of that Pair.
  5. But, if the number of pairs is even, then find both lower and upper weighted medians:
    • For the lower median traverse over the set pairs from the left and compute sum by adding weights.
    • When the sum becomes greater than or equal to 0.5 print the arr[i] value of that Pair.
    • For the upper median traverse over the set pairs from the right and compute sum by adding weights.
    • When the sum becomes greater than or equal to 0.5 print the arr[i] value of that Pair.

Below is the implementation of the above approach:

C++14

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// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate weighted median
void weightedMedian(vector<int> arr,
                    vector<float> W)
{
     
    // Store pr of arr[i] and W[i]
    vector<pair<int, float>> pr;
 
    for(int index = 0;
            index < arr.size();
            index++)
        pr.push_back({arr[index],
                        W[index]});
 
    // Sort the list of pr w.r.t.
    // to their arr[] values
    sort(pr.begin(), pr.end());
     
    // If N is odd
    if (arr.size() % 2 != 0)
    {
         
        // Traverse the set pr
        // from left to right
        float sums = 0;
        for(auto element : pr)
        {
             
            // Update sums
            sums += element.second;
 
            // If sum becomes > 0.5
            if (sums > 0.5)
                cout << "The Weighted Median is element "
                     << element.first << endl;
        }
    }
       
    // If N is even
    else
    {
         
        // For lower median traverse
        // the set pr from left
        float sums = 0;
        for(auto element : pr)
        {
             
            // Update sums
            sums += element.second;
 
            // When sum >= 0.5
            if (sums >= 0.5)
            {
                cout << "Lower Weighted Median is element "
                     << element.first << endl;
                break;
            }
        }
         
        // For upper median traverse
        // the set pr from right
        sums = 0;
        for(int index = pr.size() - 1;
                index >= 0;
                index--)
        {
            int element = pr[index].first;
            float weight = pr[index].second;
 
            // Update sums
            sums += weight;
 
            // When sum >= 0.5
            if (sums >= 0.5)
            {
                cout << "Upper Weighted Median is element "
                     << element;
                break;
            }
        }
    }
}
 
// Driver Code
int main()
{
     
    // Given array arr[]
    vector<int> arr = { 4, 1, 3, 2 };
     
    // Given weights W[]
    vector<float> W = { 0.25, 0.49, 0.25, 0.01 };
     
    // Function Call
    weightedMedian(arr, W);
}
 
// This code is contributed by mohit kumar 29

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Java

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// Java program for the
// above approach
import java.util.*;
class GFG{
 
static class Pair implements Comparable<Pair>
{
  int first;
  double second;
 
  Pair(int f, double s)
  {
    first = f;
    second = s;
  }
 
  @Override
  public int compareTo(Pair o)
  {
    if(this.second > o.second)
      return 1;
    else if(this.second == o.second)
      return 0;
    return -1;
  }
}
 
// Function to calculate weighted median
static void weightedMedian(Vector<Integer> arr,
                           Vector<Double> W)
{
  // Store pr of arr[i] and W[i]
  Vector<Pair> pr = new Vector<>();
 
  for(int index = 0;
      index < arr.size();
      index++)
    pr.add(new Pair(arr.get(index),
                    W.get(index)));
 
  // Sort the list of pr w.r.t.
  // to their arr[] values
  Collections.sort(pr);
 
  // If N is odd
  if (arr.size() % 2 != 0)
  {
    // Traverse the set pr
    // from left to right
    float sums = 0;
    for(Pair element : pr)
    {
      // Update sums
      sums += element.second;
 
      // If sum becomes > 0.5
      if (sums > 0.5)
        System.out.print(
               "The Weighted Median is element " +
                element.first + "\n");
    }
  }
 
  // If N is even
  else
  {
    // For lower median traverse
    // the set pr from left
    double sums = 0;
    for(Pair element : pr)
    {
      // Update sums
      sums += element.second;
 
      // When sum >= 0.5
      if (sums <= 0.5)
      {
        System.out.print(
               "Lower Weighted Median is element " +
                element.first + "\n");
        break;
      }
    }
 
    // For upper median traverse
    // the set pr from right
    sums = 0;
    for(int index = pr.size() - 1;
            index >= 0; index--)
    {
      int element = pr.get(index).first;
      double weight = pr.get(index).second;
 
      // Update sums
      sums += weight;
 
      // When sum >= 0.5
      if (sums >= 0.5)
      {
        System.out.print(
               "Upper Weighted Median is element " +
                element);
        break;
      }
    }
  }
}
 
// Driver Code
public static void main(String[] args)
{   
  // Given array arr[]
  Vector<Integer> arr = new Vector<>();
  arr.add(4);
  arr.add(1);
  arr.add(3);
  arr.add(2);
 
  // Given weights W[]
  Vector<Double> W =   new Vector<>();
  W.add(0.25);
  W.add(0.49);
  W.add(0.25);
  W.add(0.01);
 
  // Function Call
  weightedMedian(arr, W);
}
}
 
// This code is contributed by gauravrajput1

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Python3

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# Python3 program for the above approach
 
# Function to calculate weighted median
def weightedMedian(arr, W):
 
    # Store pairs of arr[i] and W[i]
    pairs = []
     
    for index in range(len(arr)):
        pairs.append([arr[index], W[index]])
 
    # Sort the list of pairs w.r.t.
    # to their arr[] values
    pairs.sort(key = lambda p: p[0])
 
    # If N is odd
    if len(arr) % 2 != 0:
 
        # Traverse the set pairs
        # from left to right
        sums = 0
        for element, weight in pairs:
         
            # Update sums
            sums += weight
 
            # If sum becomes > 0.5
            if sums > 0.5:
                print("The Weighted Median", end = ' ')
                print("is element {}".format(element))
 
    # If N is even
    else:
 
        # For lower median traverse
        # the set pairs from left
        sums = 0
        for element, weight in pairs:
             
            # Update sums
            sums += weight
 
            # When sum >= 0.5
            if sums >= 0.5:
                print("Lower Weighted Median", end = ' ')
                print("is element {}".format(element))
                break
 
        # For upper median traverse
        # the set pairs from right
        sums = 0
        for index in range(len(pairs)-1, -1, -1):
         
            element = pairs[index][0]
            weight = pairs[index][1]
             
            # Update sums
            sums += weight
 
            # When sum >= 0.5
            if sums >= 0.5:
                print("Upper Weighted Median", end = ' ')
                print("is element {}".format(element))
                break
 
# Driver Code
if __name__ == "__main__":
     
    # Given array arr[]
    arr = [4, 1, 3, 2]
     
    # Given weights W[]
    W = [0.25, 0.49, 0.25, 0.01]
 
    # Function Call
    weightedMedian(arr, W)

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Output: 

Lower Weighted Median is element 2
Upper Weighted Median is element 3







 

Time Complexity: O(N log N)
Auxiliary Space: O(N)

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