Minimum Cost to make all array elements equal using given operations

Given an array arr[] of positive integers and three integers A, R, M, where

  • The cost of adding 1 to an element of the array is A,
  • the cost of subtracting 1 from an element of the array is R and
  • the cost of adding 1 to an element and subtracting 1 from another element simultaneously is M.

The task is to find the minimum total cost to make all the elements of the array equal.

Examples:



Input: arr[] = {5, 5, 3, 6, 5}, A = 1, R = 2, M = 4
Output: 4
Explanation:
Operation 1: Add two times to element 3, Array – {5, 5, 5, 6, 5}, Cost = 2
Operation 2: Substract one time to element 6, Array – {5, 5, 5, 5, 5}, Cost = 4
Therefore, minimum cost is 4.

Input: arr[] = {5, 5, 3, 6, 5}, A = 1, R = 2, M = 2
Output: 3

Approach: The idea is to:

  1. find the minimum of the M and A + R as M can make both operations simultaneously.
  2. Then, store prefix sum in an array to find the sum in constant time.
  3. Now for each element calculate the cost of making equal to the current element and find the minimum of them.
  4. The smallest answer can also exist when we make all elements equal to the average of the array.
  5. Therefore, at the end also compute the cost for making all elements equal to approximate average sum of elements.

Below is the implementation of above approach:

C++

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// C++ implementation to find the
// minimum cost to make all array
// elements equal
  
#include <bits/stdc++.h>
using namespace std;
  
// Function that returns the cost of
// makeing all elements equal to current element
int costCalculation(
    int current, int arr[],
    int n, int pref[],
    int a, int r,
    int minimum)
{
  
    // Compute the lower bound
    // of current element
    int index
        = lower_bound(
              arr, arr + n, current)
          - arr;
  
    // Calculate the requirement
    // of add operation
    int left
        = index * current - pref[index];
  
    // Calcuate the requirement
    // of subtract operation
    int right
        = pref[n] - pref[index]
          - (n - index)
                * current;
  
    // Compute minimum of left and right
    int res = min(left, right);
    left -= res;
    right -= res;
  
    // Computing the total cost of add
    // and subtract operations
    int total = res * minimum;
    total += left * a;
    total += right * r;
  
    return total;
}
  
// Function that prints minimum cost
// of making all elements equal
void solve(int arr[], int n,
           int a, int r, int m)
{
    // Sort the given array
    sort(arr, arr + n);
  
    // Calcuate minimum from a + r and m
    int minimum = min(a + r, m);
  
    int pref[n + 1] = { 0 };
  
    // Compute prefix sum
    // and store in pref
    // array
    for (int i = 0; i < n; i++)
        pref[i + 1] = pref[i] + arr[i];
  
    int ans = 10000;
  
    // Find the minimum cost
    // from the given elements
    for (int i = 0; i < n; i++)
        ans
            = min(
                ans,
                costCalculation(
                    arr[i], arr, n,
                    pref, a, r,
                    minimum));
  
    // Finding the minimum cost
    // from the other cases where
    // minimum cost can occur
    ans
        = min(
            ans,
            costCalculation(
                pref[n] / n, arr,
                n, pref, a,
                r, minimum));
    ans
        = min(
            ans,
            costCalculation(
                pref[n] / n + 1,
                arr, n, pref,
                a, r, minimum));
  
    // Printing the minimum cost of making
    // all elements equal
    cout << ans << "\n";
}
  
// Driver Code
int main()
{
    int arr[] = { 5, 5, 3, 6, 5 };
  
    int A = 1, R = 2, M = 4;
  
    int size = sizeof(arr) / sizeof(arr[0]);
  
    // Function Call
    solve(arr, size, A, R, M);
  
    return 0;
}

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Java

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// Java implementation to find the
// minimum cost to make all array
// elements equal
import java.lang.*;
import java.util.*;
  
class GFG{
  
public static int lowerBound(int[] array, int length, 
                                          int value)
{
    int low = 0;
    int high = length;
      
    while (low < high)
    {
        final int mid = (low + high) / 2;
          
        // Checks if the value is less 
        // than middle element of the array
        if (value <= array[mid])
        {
            high = mid;
        }
        else
        {
            low = mid + 1;
        }
    }
    return low;
}
  
// Function that returns the cost of makeing 
// all elements equal to current element
public static int costCalculation(int current, int arr[],
                                  int n, int pref[],
                                  int a, int r, 
                                  int minimum)
{
  
    // Compute the lower bound
    // of current element
    int index = lowerBound(arr, arr.length, current);
  
    // Calculate the requirement
    // of add operation
    int left = index * current - pref[index];
  
    // Calcuate the requirement
    // of subtract operation
    int right = pref[n] - 
                pref[index]- (n - index)* current;
  
    // Compute minimum of left and right
    int res = Math.min(left, right);
    left -= res;
    right -= res;
  
    // Computing the total cost of add
    // and subtract operations
    int total = res * minimum;
    total += left * a;
    total += right * r;
  
    return total;
}
  
// Function that prints minimum cost
// of making all elements equal
public static void solve(int arr[], int n,
                         int a, int r, int m)
{
      
    // Sort the given array
    Arrays.sort(arr);
  
    // Calcuate minimum from a + r and m
    int minimum = Math.min(a + r, m);
  
    int []pref = new int [n + 1]; 
    Arrays.fill(pref, 0);
      
    // Compute prefix sum and 
    // store in pref array
    for(int i = 0; i < n; i++)
       pref[i + 1] = pref[i] + arr[i];
  
    int ans = 10000;
  
    // Find the minimum cost
    // from the given elements
    for(int i = 0; i < n; i++)
       ans = Math.min(ans, costCalculation(arr[i], arr, 
                                           n, pref, 
                                           a, r, minimum));
  
    // Finding the minimum cost
    // from the other cases where
    // minimum cost can occur
    ans = Math.min(ans, costCalculation(pref[n] / n, arr,
                                        n, pref, a, r,
                                        minimum));
    ans = Math.min(ans, costCalculation(pref[n] / n + 1
                                        arr, n, pref,
                                        a, r, minimum));
  
    // Printing the minimum cost of making
    // all elements equal
    System.out.println(ans);
}
  
// Driver Code
public static void main(String args[])
{
    int arr[] = { 5, 5, 3, 6, 5 };
    int A = 1, R = 2, M = 4;
    int size = arr.length ;
  
    // Function Call
    solve(arr, size, A, R, M);
}
}
  
// This code is contributed by SoumikMondal

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Output:

4

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Improved By : SoumikMondal