Make array elements equal in Minimum Steps

Given an array of N elements where the first element is a non zero positive number M, and the rest N – 1 elements are 0, the task is to calculate the minimum number of steps required to make the entire array equal while abiding by the following rules:

1. The ith element can be increased by one if and only if i-1th element is strictly greater than the ith element
2. If the ith element is being increased by one then the i+1th cannot be increased at the same time.(i.e consecutive elements cannot be increased at the same time)
3. Multiple elements can be incremented simultaneously in a single step.

Examples:

Input : N = 3, M = 4
Output : 8
Explanation:
array is 4 0 0
In 4 steps element at index 1 is increased, so the array becomes {4, 4, 0}. In the next 4 steps the element at index 3 is increased so array becomes {4, 4, 4}
Thus, 4 + 4 = 8 operations are required to make all the array elements equal

Input : N = 4, M = 4
Output : 9
Explanation:
The steps are shown in the flowchart given below
Refer to the flowchart given below.

Flowchart for output

Approach:
To maximise the Number of Increments per Step, more number of Unbalances are created (array[i]>array[i+1]),

Step 1, element 0 >element 1 so element 1 is incremented,

Step 2, element 1> element 2 so element 2 is incremented by 1

Step 3, element 0 > element 1 and element 2> element 3 so element 1 &3 are incremented by 1

Step 4, element 1 > element 2 element 3 > element 4 so element 2 & 4 are incremented

Step 5, element 0> element 1; element 2>element 3 ;element 4> element 5; so element 1, 3, &5 are incremented.
and so on…

Consider the following array,
5 0 0 0 0 0

1) 5 1 0 0 0 0
2) 5 1 1 0 0 0
3) 5 2 1 1 0 0
4) 5 2 2 1 1 0
5) 5 3 2 2 1 1
6) 5 3 3 2 2 1
7) 5 4 3 3 2 2
8) 5 4 4 3 3 2
9) 5 5 4 4 3 3
10) 5 5 5 4 4 3
11) 5 5 5 5 4 4
12) 5 5 5 5 5 4
13) 5 5 5 5 5 5

Notice that after an unbalance is created (i.e array[i]>array[i+1]) the element gets incremented by one in alternate steps. In step 1 element 1 gets incremented to 1, in step 2 element 2 gets incremented to 1, in step 3 element 3 gets incremented to 1, so in step n-1, n-1th element will become 1. After that n-1th element is increased by 1 on alternate steps until it reaches the value at element 0. Then the entire array becomes equal.

So the pattern followed by the last element is
(0, 0, 0.., 0) till (N – 4)th element becomes 1 which is n-4 steps
and after that,
(0, 0, 1, 1, 2, 2, 3, 3, 4, 4, … M – 1, M – 1, M) which is 2*m + 1 steps.

So the Final Result becomes (N – 3) + 2 * M

There are a few corner cases which need to be handled, viz. When N = 1, array has only a single element, so the number of steps required = 0. and When N = 2, number of steps required equal to M

C++

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// C++ program to make the array elements equal in minimum steps
  
#include <bits/stdc++.h>
using namespace std;
  
// Returns the minumum steps required to make an array of N
// elements equal, where the first array element equals M
int steps(int N, int M)
{
    // Corner Case 1: When N = 1
    if (N == 1)
        return 0;
  
    // Corner Case 2: When N = 2
    else if (N == 2) // corner case 2
        return M;
  
    return 2 * M + (N - 3);
}
  
// Driver Code
int main()
{
    int N = 4, M = 4;
    cout << steps(N, M);
    return 0;
}

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Java

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// Java program to make the array elements
// equal in minimum steps
  
import java.io.*;
  
class GFG {
  
    // Returns the minumum steps required
    // to make an array of N elements equal,
    // where the first array element equals M
    static int steps(int N, int M)
    {
        // Corner Case 1: When N = 1
        if (N == 1)
            return 0;
      
        // Corner Case 2: When N = 2
        else if (N == 2) // corner case 2
            return M;
      
        return 2 * M + (N - 3);
    }
      
    // Driver Code
    public static void main (String[] args) 
    {
        int N = 4, M = 4;
        System.out.print( steps(N, M));
    }
}
  
// This code is contributed by anuj_67.

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Python3

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# Python program to make 
# the array elements equal
# in minimum steps
  
# Returns the minumum steps 
# required to make an array 
# of N elements equal, where
# the first array element 
# equals M
def steps(N, M):
  
    # Corner Case 1: When N = 1
    if (N == 1):
        return 0
  
    # Corner Case 2: When N = 2
    elif(N == 2): 
        return M
  
    return 2 * M + (N - 3)
  
# Driver Code
N = 4
M = 4
print(steps(N,M))
  
# This code is contributed
# by Shivi_Aggarwal.

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C#

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// C# program to make the array 
// elements equal in minimum steps
using System;
  
class GFG
{
  
    // Returns the minumum steps 
    // required to make an array 
    // of N elements equal, where 
    // the first array element 
    // equals M
    static int steps(int N, int M)
    {
        // Corner Case 1: When N = 1
        if (N == 1)
             return 0;
      
        // Corner Case 2: When N = 2
        else if (N == 2) // corner case 2
            return M;
      
        return 2 * M + (N - 3);
    }
      
    // Driver Code
    public static void Main () 
    {
        int N = 4, M = 4;
        Console.WriteLine(steps(N, M));
    }
}
  
// This code is contributed by anuj_67.

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PHP

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<?php
// PHP program to make the array 
// elements equal in minimum steps 
  
// Returns the minumum steps required 
// to make an array of N elements equal,
// where the first array element equals M 
function steps($N, $M
    // Corner Case 1: When N = 1 
    if ($N == 1) 
        return 0; 
  
    // Corner Case 2: When N = 2 
    else if ($N == 2) // corner case 2 
        return $M
  
    return 2 * $M + ($N - 3); 
  
// Driver Code 
$N = 4;
$M = 4; 
echo steps($N, $M); 
      
// This code is contributed by ajit
?>

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Output:

9


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Improved By : vt_m, Shivi_Aggarwal, jit_t



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