Minimum number of steps required to obtain the given Array by the given operations

Given an array arr[] of N positive integers, the task is to find the minimum number of operations required of the following types to obtain the array arr[] from an array of zeroes only.

  • Select any index i and increment all the elements at the indices [i, N – 1] by 1.
  • Select any index i and decrease all the elements at the indices [i, N – 1] by 1.

Examples: 
 

Input: arr[]={1,1,2,2,1} 
Output:
Explanation: 
Initially arr[] = {0,0,0,0,0} 
Step 1: arr[]={1,1,1,1,1}(Operation 1 on index 0) 
Step 2: arr[]={1,1,2,2,2}(Operation 1 on index 2) 
Step 3: arr[]={1,1,2,2,1}(Operation 2 on index 4)
Input: arr[]={1,2,3,4} 
Output:
 

Naive Approach: The simplest approach is to convert each element of the array result array, brr[] to arr[] by performing one of the above operations over the indices [i, N – 1] and increase the count of each operation performed. 
Time Complexity: O(N2
Auxiliary Space: O(1)
Efficient Approach: 
The above approach can be optimized using Greedy Approach
Follow the steps below to solve the problem: 
 

  • For the 0th index, convert the number 0 to arr[0]. Therefore, the number of steps required will always be a[0]. Hence, add arr[0] to the answer.
  • For all the other indices, the common greedy observation is to use the increase or the decrease operation by abs(a[i]-a[i-1]) times.
  • The intuition behind this approach is, if the number is less than a[i-1], then increase everything from ((i-1).. n-1) by a[i], and then decrease (a[i-1] – a[i]) to get a[i].
  • If a[i] > a[i-1], the approach is to use the increase operation from ((i-1)..n-1) by a[i-1], and for the remaining value, we increase it by (a[i]-a[i-1]) operations from (i..(n-1)).
  • Therefore, traverse the array and for every element after the first, add the absolute difference of consecutive pairs to the answer.
  • Finally, print the answer.

Below is the implementation of the above approach:
 



C++

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// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the minimum
// steps to obtain the desired array
int min_operation(int a[], int n)
{
    // Initialize variable
    int ans = 0;
 
    // Iterate over the array arr[]
    for (int i = 0; i < n; i++) {
 
        // Check if i > 0
        if (i > 0)
 
            // Update the answer
            ans += abs(a[i] - a[i - 1]);
 
        else
            ans += abs(a[i]);
    }
 
    // Return the result
    return ans;
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    cout << min_operation(arr, n);
 
    return 0;
}

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Java

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// Java Program to implement
// the above approach
import java.util.*;
class GFG{
 
// Function to calculate the minimum
// steps to obtain the desired array
static int min_operation(int a[], int n)
{
    // Initialize variable
    int ans = 0;
 
    // Iterate over the array arr[]
    for (int i = 0; i < n; i++)
    {
 
        // Check if i > 0
        if (i > 0)
 
            // Update the answer
            ans += Math.abs(a[i] - a[i - 1]);
 
        else
            ans += Math.abs(a[i]);
    }
 
    // Return the result
    return ans;
}
 
// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 3, 4 };
    int n = arr.length;
 
    System.out.print(min_operation(arr, n));
}
}
 
// This code is contributed by gauravrajput1

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Python3

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# Python3 program to implement
# the above approach
 
# Function to calculate the minimum
# steps to obtain the desired array
def min_operation(a, n):
 
    # Initialize variable
    ans = 0
 
    # Iterate over the array arr[]
    for i in range(n):
 
        # Check if i > 0
        if (i > 0):
 
            # Update the answer
            ans += abs(a[i] - a[i - 1])
        else:
            ans += abs(a[i])
 
    # Return the result
    return ans
 
# Driver Code
if __name__ == "__main__":
 
    arr = [ 1, 2, 3, 4 ]
    n = len(arr)
 
    print(min_operation(arr, n))
 
# This code is contributed by chitranayal

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C#

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// C# Program to implement
// the above approach
using System;
class GFG{
 
// Function to calculate the minimum
// steps to obtain the desired array
static int min_operation(int []a, int n)
{
    // Initialize variable
    int ans = 0;
 
    // Iterate over the array []arr
    for (int i = 0; i < n; i++)
    {
 
        // Check if i > 0
        if (i > 0)
 
            // Update the answer
            ans += Math.Abs(a[i] - a[i - 1]);
 
        else
            ans += Math.Abs(a[i]);
    }
 
    // Return the result
    return ans;
}
 
// Driver Code
public static void Main(String[] args)
{
    int []arr = { 1, 2, 3, 4 };
    int n = arr.Length;
 
    Console.Write(min_operation(arr, n));
}
}
 
// This code is contributed by Amit Katiyar

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Output: 
 

4



 

Time Complexity: O(N)
Auxiliary Space: O(1)

 

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