Minimize the sum calculated by repeatedly removing any two elements and inserting their sum to the Array

Given N elements, you can remove any two elements from the list, note their sum and add the sum to the list. Repeat these steps while there are more than a single element in the list. The task is to minimize the sum of these chosen sum in the end.

Examples:

Input: arr[] = {1, 4, 7, 10}
Output: 39
Choose 1 and 4, Sum = 5, arr[] = {5, 7, 10}
Choose 5 and 7, Sum = 17, arr[] = {12, 10}
Choose 12 and 10, Sum = 39, arr[] = {22}

Input: arr[] = {1, 3, 7, 5, 6}
Output: 48



Approach: In order to minimize the sum, the elements that gets chosen at every step must the minimum elements from the list. In order to do that efficiently, a priority queue can be used. At every step, while there are more than a single element in the list, choose the minimum and the second minimum, remove them from the list add their sum to the list after updating the running sum.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include<bits/stdc++.h>
using namespace std;
  
// Function to return the minimized sum
int getMinSum(int arr[], int n)
{
    int i, sum = 0;
  
    // Priority queue to store the elements of the array
    // and retrieve the minimum element efficiently
    priority_queue<int, vector<int>, greater<int> > pq;
  
    // Add all the elements
    // to the priority queue
    for (i = 0; i < n; i++)
        pq.push(arr[i]);
  
    // While there are more than 1 elements
    // left in the queue
    while (pq.size() > 1) 
    {
  
        // Remove and get the minimum
        // element from the queue
        int min = pq.top();
  
        pq.pop();
  
        // Remove and get the second minimum
        // element (currently minimum)
        int secondMin = pq.top();
          
        pq.pop();
  
        // Update the sum
        sum += (min + secondMin);
  
        // Add the sum of the minimum
        // elements to the queue
        pq.push(min + secondMin);
    }
  
    // Return the minimized sum
    return sum;
}
  
// Driver code
int main()
{
  
    int arr[] = { 1, 3, 7, 5, 6 };
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << (getMinSum(arr, n));
}
  
// This code is contributed by mohit

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Java

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// Java implementation of the approach
import java.util.PriorityQueue;
  
class GFG
{
  
    // Function to return the minimized sum
    static int getMinSum(int arr[], int n)
    {
        int i, sum = 0;
  
        // Priority queue to store the elements of the array
        // and retrieve the minimum element efficiently
        PriorityQueue<Integer> pq = new PriorityQueue<>();
  
        // Add all the elements
        // to the prioriry queue
        for (i = 0; i < n; i++)
            pq.add(arr[i]);
  
        // While there are more than 1 elements
        // left in the queue
        while (pq.size() > 1)
        {
  
            // Remove and get the minimum
            // element from the queue
            int min = pq.poll();
  
            // Remove and get the second minimum
            // element (currently minimum)
            int secondMin = pq.poll();
  
            // Update the sum
            sum += (min + secondMin);
  
            // Add the sum of the minimum
            // elements to the queue
            pq.add(min + secondMin);
        }
  
        // Return the minimized sum
        return sum;
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 1, 3, 7, 5, 6 };
        int n = arr.length;
        System.out.print(getMinSum(arr, n));
    }
}

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Output:

48


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