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++ 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);
}

// 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

# 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

// 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

<?php
// PHP implementation of the approach

// Function to return the minimum cost
function minCost($n, $arr, $cost)
{
    $sum = 0;
    $totalCost = 0;

    // Sum of all the array elements
    // except the last element
    for ($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
$arr = array( 1, 2, 4, 5 );
$n = sizeof($arr);
$cost = 1;
echo minCost($n, $arr, $cost);

// This code is contributed by ajit
?>

<script>
    // Javascript implementation of the approach 
    
    // Function to return the minimum cost 
    function minCost(n, arr, cost) 
    { 
        let sum = 0, totalCost = 0; 
      
        // Sum of all the array elements 
        // except the last element 
        for (let 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; 
    } 
    
    let arr = [ 1, 2, 4, 5 ]; 
    let n = arr.length; 
    let cost = 1; 
    document.write(minCost(n, arr, cost)); 
    
</script>

31

Complexity Analysis:

• Time Complexity: O(n)
• Auxiliary Space: O(1)