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:

Python3

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# Python 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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