Minimum number of operations on an array to make all elements 0

Given an array arr[] of N integers and an integer cost, the task is to calculate the cost of making all the elements of the array 0 with the given operation. In a single operation, an index 0 ≤ i < N and an integer X > 0 can be chosen such that 0 ≤ i + X < N then elements can be updated as arr[i] = arr[i] – 1 and arr[i + X] = arr[i + X] + 1. If i + X ≥ N then only arr[i] will be updated but with twice the regular cost. Print the minimum cost required.

Examples:

Input: arr[] = {1, 2, 4, 5}, cost = 1
Output: 31
Move 1: i = 0, X = 3, arr[] = {0, 2, 4, 6} (cost = 1)
Moves 2 and 3: i = 1, X = 2, arr[] = {0, 0, 4, 8} (cost = 2)
Moves 4, 5, 6 and 7: i = 2, X = 1, arr[] = {0, 0, 0, 12} (cost = 4)
Move 8: i = 3, X > 0, arr[] = {0, 0, 0, 0} (cost = 24)
Total cost = 1 + 2 + 4 + 24 = 31

Input: arr[] = {1, 1, 0, 5}, cost = 2
Output: 32

Approach: To minimize the cost, for every index i always choose X such that i + X = N – 1 i.e. the last element then minimum cost can be calculated as:

  • Store the sum of the elements from arr[0] to arr[n – 2] in sum then update totalCost = cost * sum and arr[n – 1] = arr[n – 1] + sum.
  • Now the cost of making all the elements 0 except the last one has been calculated. And the cost of making the last element 0 can be calculated as totalCost = totalCost + (2 * cost * arr[n – 1]).

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 minimum cost
int minCost(int n, int arr[], int cost)
{
    int sum = 0, totalCost = 0;
  
    // Sum of all the array elements
    // except the last element
    for (int i = 0; i < n - 1; i++)
        sum += arr[i];
  
    // Cost of making all the array elements 0
    // except the last element
    totalCost += cost * sum;
  
    // Update the last element
    arr[n - 1] += sum;
  
    // Cost of making the last element 0
    totalCost += (2 * cost * arr[n - 1]);
  
    return totalCost;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 2, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int cost = 1;
    cout << minCost(n, arr, cost);
}

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Java

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// Java implementation of the approach 
public class GfG
{
  
    // Function to return the minimum cost 
    static int minCost(int n, int arr[], int cost) 
    
        int sum = 0, totalCost = 0
      
        // Sum of all the array elements 
        // except the last element 
        for (int i = 0; i < n - 1; i++) 
            sum += arr[i]; 
      
        // Cost of making all the array elements 0 
        // except the last element 
        totalCost += cost * sum; 
      
        // Update the last element 
        arr[n - 1] += sum; 
      
        // Cost of making the last element 0 
        totalCost += (2 * cost * arr[n - 1]); 
      
        return totalCost; 
    
  
    // Driver code 
    public static void main(String []args)
    {
          
        int arr[] = { 1, 2, 4, 5 }; 
        int n = arr.length; 
        int cost = 1
        System.out.println(minCost(n, arr, cost));
    }
}
  
// This code is contributed by Rituraj Jain

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Python3

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# Python3 implementation of the approach 
  
# Function to return the minimum cost 
def minCost(n, arr, cost): 
  
    Sum, totalCost = 0, 0
  
    # Sum of all the array elements 
    # except the last element 
    for i in range(0, n - 1): 
        Sum += arr[i] 
  
    # Cost of making all the array elements 0 
    # except the last element 
    totalCost += cost * Sum
  
    # Update the last element 
    arr[n - 1] += Sum
  
    # Cost of making the last element 0 
    totalCost += (2 * cost * arr[n - 1]) 
  
    return totalCost 
  
# Driver code 
if __name__ == "__main__":
  
    arr = [1, 2, 4, 5
    n = len(arr) 
    cost = 1
    print(minCost(n, arr, cost)) 
  
# This code is contributed by Rituraj Jain

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C#

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// C# implementation of the approach 
using System ;
  
class GfG 
  
    // Function to return the minimum cost 
    static int minCost(int n, int []arr, int cost) 
    
        int sum = 0, totalCost = 0; 
      
        // Sum of all the array elements 
        // except the last element 
        for (int i = 0; i < n - 1; i++) 
            sum += arr[i]; 
      
        // Cost of making all the array elements 0 
        // except the last element 
        totalCost += cost * sum; 
      
        // Update the last element 
        arr[n - 1] += sum; 
      
        // Cost of making the last element 0 
        totalCost += (2 * cost * arr[n - 1]); 
      
        return totalCost; 
    
  
    // Driver code 
    public static void Main() 
    
          
        int []arr = { 1, 2, 4, 5 }; 
        int n = arr.Length; 
        int cost = 1; 
        Console.WriteLine(minCost(n, arr, cost)); 
    
  
// This code is contributed by Ryuga

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Output:

31

Time Complexity: O(n)



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