Minimum sum after subtracting multiples of k from the elements of the array

Given an integer and an integer array, the task is to find the minimum possible sum of all the elements of the array after they are reduced by subtracting a multiple of from each element (the result must be positive and every element of the array must be equal after this reduction). If the array cannot be reduced then print 

Note that an element may or may not be reduced in the final state of the array.

Examples:

Input: arr[] = {2, 3, 4, 5}, K = 1 
Output: 4 
Subtract 1 form 2, arr[] = {1, 3, 4, 5} 
Subtract 2 from 3, arr[] = {1, 1, 4, 5} 
Subtract 3 from 4, arr[] = {1, 1, 1, 5} 
Subtract 4 from 5 to make arr[] = {1, 1, 1, 1}, thus giving minimum possible sum as 4.

Input: arr[] = {5, 6, 7}, K = 2 
Output: -1 

Approach: First, the array needs to be sorted as the problem can be solved using the greedy approach. 

• Sort the array, if arr[0] < 0 then print -1 as every element needs to be ? 0.
• If K == 0 then no element can be reduced further. So in order to have an answer every element of the array must be equal. So the sum of elements is n * arr[0] else print -1.
• Now for the rest of the elements, run a loop from 1 to n and check whether ((arr[i] – arr[0]) % K) == 0 i.e. arr[i] can be reduced to arr[0].
• If the above condition fails for any element, print -1.
• Else if k == 1 then the answer is n i.e. every element will get reduced to 1.
• Else the answer is n * (a[0] % k).

Below is the implementation of the above approach:  

C++

// C++ program of the above approach
#include <bits/stdc++.h>
using namespace std;

// function to calculate minimum sum after transformation
int min_sum(int n, int k, int a[])
{

    sort(a, a + n);

    if (a[0] < 0)
        return -1;


    // no element can be reduced further
    if (k == 0) {

        // if all the elements of the array
        // are identical
        if (a[0] == a[n - 1])
            return (n * a[0]);
        else
            return -1;
    }
    else {
        int f = 0;
        for (int i = 1; i < n; i++) {

            int p = a[i] - a[0];

            // check if a[i] can be reduced to a[0]
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (a[0] % k));
        }
    }
}

// Driver code
int main()
{
    int arr[] = { 2, 3, 4, 5 };
    int K = 1;
    int N = sizeof(arr) / sizeof(arr[0]);
    cout << min_sum(N, K, arr);

    return 0;
}

Java

// Java program of the above approach

import java.io.*;
import java.util.*;
class GFG {

// function to calculate minimum sum after transformation
static int min_sum(int n, int k, int a[])
{

    Arrays.sort(a);

    if (a[0] < 0)
        return -1;


    // no element can be reduced further
    if (k == 0) {

        // if all the elements of the array
        // are identical
        if (a[0] == a[n - 1])
            return (n * a[0]);
        else
            return -1;
    }
    else {
        int f = 0;
        for (int i = 1; i < n; i++) {

            int p = a[i] - a[0];

            // check if a[i] can be reduced to a[0]
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f>0)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (a[0] % k));
        }
    }
}

// Driver code


    public static void main (String[] args) {
            int arr[] = { 2, 3, 4, 5 };
    int K = 1;
    int N = arr.length;
    System.out.println(min_sum(N, K, arr));
    }
}
// This code is contributed by shs..

Python3

# Python 3 program of the above approach

# function to calculate minimum sum
# after transformation
def min_sum(n, k, a):
    a.sort(reverse = False)

    if (a[0] < 0):
        return -1

    # no element can be reduced further
    if (k == 0):

        # if all the elements of the
        # array are identical
        if (a[0] == a[n - 1]):
            return (n * a[0])
        else:
            return -1

    else:
        f = 0
        for i in range(1, n, 1):
            p = a[i] - a[0]

            # check if a[i] can be
            # reduced to a[0]
            if (p % k == 0):
                continue
            else:
                f = 1
                break

        # one of the elements cannot be reduced
        # to be equal to the other elements
        if (f):
            return -1
        else:

            # if k = 1 then all elements
            # can be reduced to 1
            if (k == 1):
                return n
            else:
                return (n * (a[0] % k))

# Driver code
if __name__ == '__main__':
    arr = [2, 3, 4, 5]
    K = 1
    N = len(arr)
    print(min_sum(N, K, arr))

# This code is contributed by
# Surendra_Gangwar

C#

// C# program of the above approach
using System;

class GFG
{

// function to calculate minimum
// sum after transformation
static int min_sum(int n, int k, int[] a)
{

    Array.Sort(a);

    if (a[0] < 0)
        return -1;

    // no element can be reduced further
    if (k == 0)
    {

        // if all the elements of the array
        // are identical
        if (a[0] == a[n - 1])
            return (n * a[0]);
        else
            return -1;
    }
    else
    {
        int f = 0;
        for (int i = 1; i < n; i++)
        {
            int p = a[i] - a[0];

            // check if a[i] can be
            // reduced to a[0]
            if (p % k == 0)
                continue;
            else
            {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f > 0)
            return -1;
        else
        {

            // if k = 1 then all elements can
            // be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (a[0] % k));
        }
    }
}

// Driver code
public static void Main ()
{
    int[] arr = new int[] { 2, 3, 4, 5 };
    int K = 1;
    int N = arr.Length;
    Console.WriteLine(min_sum(N, K, arr));
}
}

// This code is contributed by mits

PHP

<?php
// PHP program of the above approach

// function to calculate minimum
// sum after transformation
function min_sum($n, $k, $a)
{

    sort($a);

    if ($a[0] < 0)
        return -1;


    // no element can be reduced further
    if ($k == 0)
    {

        // if all the elements of the array
        // are identical
        if ($a[0] == $a[$n - 1])
            return ($n * $a[0]);
        else
            return -1;
    }
    else
    {
        $f = 0;
        for ($i = 1; $i <$n; $i++)
        {

            $p = $a[$i] - $a[0];

            // check if a[i] can be reduced to a[0]
            if ($p % $k == 0)
                continue;
            else
            {
                $f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if ($f)
            return -1;
        else
        {

            // if k = 1 then all elements can
            // be reduced to 1
            if ($k == 1)
                return $n;
            else
                return ($n * ($a[0] % $k));
        }
    }
}

// Driver code
$arr = array(2, 3, 4, 5 );
$K = 1;
$N = count($arr);
echo min_sum($N, $K, $arr);

// This code is contributed by inder_verma
?>

Javascript

<script>

// Javascript program of the above approach

// function to calculate minimum sum after transformation
function min_sum(n, k, a)
{

    a.sort();

    if (a[0] < 0)
        return -1;


    // no element can be reduced further
    if (k == 0) {

        // if all the elements of the array
        // are identical
        if (a[0] == a[n - 1])
            return (n * a[0]);
        else
            return -1;
    }
    else {
        let f = 0;
        for (let i = 1; i < n; i++) {

            let p = a[i] - a[0];

            // check if a[i] can be reduced to a[0]
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f>0)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (a[0] % k));
        }
    }
}

// Driver code

    let arr = [ 2, 3, 4, 5 ];
    let K = 1;
    let N = arr.length;
    document.write(min_sum(N, K, arr));

</script>

Output:

4

Time Complexity: O(nlogn)

Auxiliary Space: O(1)

Further Optimizations: 

Instead of sorting the array, we can find the minimum element in O(n) time. We can check whether all elements are the same or not also in O(n) time.

The steps involved in this approach are as follows:

1.  Find the minimum element in the array and store it in variable name val.
2.  Now if val < 0 then print -1 as every element needs to be ? 0 and if the minimum element is greater or equal to 0 then every element will be greater than 0.
3.  If K == 0 then no element can be reduced further. So in order to have an answer every element of the array must be equal. So we check this by iterating a loop and if all are equal then the answer will be n * val else print -1.
4.  Now for the rest of the elements, iterate a loop and check whether ((arr[i] – val) % K) == 0 i.e.  arr[i] can be reduced to val.
5.  If the above condition fails for any element, print -1.
6.  Else if k == 1 then the answer is n i.e. every element will get reduced to 1.
7.  Else the answer is n * (val % k).

Below is the code for the above approach:

C++

// C++ program of the above approach

#include <bits/stdc++.h>

using namespace std;

// function to calculate minimum sum after transformation
int min_sum(int n, int k, int a[])
{
    // Finding minimum element in the array
    int val=INT_MAX;
    for(int i=0;i<n;i++)
    {
        val=min(a[i],val);
    }

    if (val < 0)
        return -1;

    // no element can be reduced further
    if (k == 0) {

        // check if all the elements of the array
        // are identical or not
        for(int i=0;i<n;i++)
        {
            if(val!=a[i])
            return -1;
        }
            return (n * val);
    }
    else {
        int f = 0;
        for (int i = 0; i < n; i++) {

            int p = a[i] - val;

            // check if a[i] can be reduced to val
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (val % k));
        }
    }
}

// Driver code
int main()
{
    int arr[] = { 2, 3, 4, 5 };
    int K = 1;
    int N = sizeof(arr) / sizeof(arr[0]);
    cout << min_sum(N, K, arr);

    return 0;
}

// This code is contributed by Pushpesh raj

Java

// Java program of above approach
import java.io.*;
import java.util.*;

class GFG{

static int min_sum(int n,int k,int[] a)
{
    // Finding minimum element in the array
    int val=Integer.MAX_VALUE;
    for(int i=0;i<n;i++)
    {
        val=Math.min(a[i],val);
    }

    if (val < 0)
        return -1;

    // no element can be reduced further
    if (k == 0) {

        // check if all the elements of the array
        // are identical or not
        for(int i=0;i<n;i++)
        {
            if(val!=a[i])
            return -1;
        }
            return (n * val);
    }
    else {
        int f = 0;
        for (int i = 0; i < n; i++) {

            int p = a[i] - val;

            // check if a[i] can be reduced to val
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f!=0)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (val % k));
        }
    }
}

// Driver Code
public static void main(String[] args)
{
    int[] arr = { 2, 3, 4, 5 };
    int K = 1;
    int N = arr.length;

    System.out.println(min_sum(N, K, arr));
}
}

// This code is contributed by aditya942003patil

Python3

# Python3 program of the above approach

# function to calculate minimum sum after transformation
def min_sum(n, k, a):

    # Finding minimum element in the array
    val = 10000000;
    for i in range(n):

        val = min(a[i], val);


    if (val < 0):
        return -1;

    # no element can be reduced further
    if (k == 0) :

        # check if all the elements of the array
        # are identical or not
        for i in range(n):

            if(val != a[i]):
                return -1;

            return (n * val);

    else :
        f = 0;
        for i in range(n):

            p = a[i] - val;

            # check if a[i] can be reduced to val
            if (p % k == 0):
                continue;
            else :
                f = 1;
                break;



        # one of the elements cannot be reduced
        # to be equal to the other elements
        if (f > 0):
            return -1;
        else :

            # if k = 1 then all elements can be reduced to 1
            if (k == 1):
                return n;
            else:
                return (n * (val % k));

# Driver code
arr = [ 2, 3, 4, 5 ];
K = 1;
N = len(arr);
print(min_sum(N, K, arr));


# This code is contributed by phasing17

C#

// C# program of the above approach
using System;
class GFG{

static int min_sum(int n, int k, int[] a)
{

    // Finding minimum element in the array
    int val = Int32.MaxValue;
    for(int i = 0; i < n; i++)
    {
        val = Math.Min(a[i],val);
    }

    if (val < 0)
        return -1;

    // no element can be reduced further
    if (k == 0) {

        // check if all the elements of the array
        // are identical or not
        for(int i = 0; i < n; i++)
        {
            if(val !=
               a[i])
            return -1;
        }
            return (n * val);
    }
    else {
        int f = 0;
        for (int i = 0; i < n; i++) {

            int p = a[i] - val;

            // check if a[i] can be reduced to val
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f!=0)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (val % k));
        }
    }
}

// Driver Code
public static void Main ()
{
    int[] arr = { 2, 3, 4, 5 };
    int K = 1;
    int N = arr.Length;

    Console.WriteLine(min_sum(N, K, arr));
}
}

// This code is contributed by Aman Kumar

Javascript

// JavaScript program of the above approach

// function to calculate minimum sum after transformation
function min_sum(n, k, a)
{
    // Finding minimum element in the array
    let val = 10000000;
    for(var i = 0; i < n; i++)
    {
        val = Math.min(a[i], val);
    }

    if (val < 0)
        return -1;

    // no element can be reduced further
    if (k == 0) {

        // check if all the elements of the array
        // are identical or not
        for(var i = 0; i < n; i++)
        {
            if(val != a[i])
            return -1;
        }
            return (n * val);
    }
    else {
        var f = 0;
        for (var i = 0; i < n; i++) {

            var p = a[i] - val;

            // check if a[i] can be reduced to val
            if (p % k == 0)
                continue;
            else {
                f = 1;
                break;
            }
        }

        // one of the elements cannot be reduced
        // to be equal to the other elements
        if (f > 0)
            return -1;
        else {

            // if k = 1 then all elements can be reduced to 1
            if (k == 1)
                return n;
            else
                return (n * (val % k));
        }
    }
}

// Driver code
let arr = [ 2, 3, 4, 5 ];
let K = 1;
let N = arr.length;
console.log(min_sum(N, K, arr));

// This code is contributed by phasing17

Output

4

Time Complexity: O(n)

Auxiliary Space: O(1)