Smallest positive integer K such that all array elements can be made equal by incrementing or decrementing by at most K

Given an array arr[] of size N, the task is to find the smallest positive integer K such that incrementing or decrementing each array element by K at most once makes all elements equal. If it is not possible to make all array elements equal, then print -1.

Examples :

Input: arr[] = { 5, 7, 9 }
Output: 2
Explanation:
Incrementing the value of arr[0] by K(= 2) modifies arr[] to { 7, 7, 9 }. 
Decrementing the value of arr[2] by K(= 2) modifies arr[] to { 7, 7, 7 }

Input: arr[] = {1, 3, 9}, N = 3
Output: -1

Approach: Follow the steps below to solve the problem :

• Initialize a Set to store all distinct elements present in the array.
• Count the distinct elements in the array, which is equal to the size of the Set, say M.
• If M > 3, then print -1.
• If M = 3, then check if the difference between the largest and the second largest element of the Set is equal to the difference between the second largest and the third largest element of the Set or not. If found to be true, then print the difference. Otherwise, print -1.
• If M = 2, then check if the difference between the largest and the second largest element of the set is even or not. If found to be true, then print the half of their difference. Otherwise, print the difference.
• If M <= 1, then print 0.

Below is the implementation of the above solution :

C++

// C++ program for the above approach
#include <iostream>
#include <set>
using namespace std;

// Function to find smallest integer K such that
// incrementing or decrementing each element by
// K at most once makes all elements equal
void findMinKToMakeAllEqual(int N, int A[])
{

    // Store distinct
    // array elements
    set<int> B;

    // Traverse the array, A[]
    for (int i = 0; i < N; i++)
        B.insert(A[i]);

    // Count elements into the set
    int M = B.size();

    // Iterator to store first
    // element of B
    set<int>::iterator itr = B.begin();

    // If M is greater than 3
    if (M > 3)
        printf("-1");

    // If M is equal to 3
    else if (M == 3) {

        // Stores the first
        // smallest element
        int B_1 = *itr;

        // Stores the second
        // smallest element
        int B_2 = *(++itr);

        // Stores the largest element
        int B_3 = *(++itr);

        // IF difference between B_2 and B_1
        // is equal to B_3 and B_2
        if (B_2 - B_1 == B_3 - B_2)
            printf("%d", B_2 - B_1);
        else
            printf("-1");
    }

    // If M is equal to 2
    else if (M == 2) {

        // Stores the smallest element
        int B_1 = *itr;

        // Stores the largest element
        int B_2 = *(++itr);

        // If difference is an even
        if ((B_2 - B_1) % 2 == 0)
            printf("%d", (B_2 - B_1) / 2);
        else
            printf("%d", B_2 - B_1);
    }

    // If M is equal to 1
    else
        printf("%d", 0);
}

// Driver Code
int main()
{

    // Given array
    int A[] = { 1, 3, 5, 1 };

    // Given size
    int N = sizeof(A) / sizeof(A[0]);

    // Print the required smallest integer
    findMinKToMakeAllEqual(N, A);
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;
class GFG {

    // Function to find smallest integer K such that
    // incrementing or decrementing each element by
    // K at most once makes all elements equal
    static void findMinKToMakeAllEqual(int N, int A[])
    {

        // Store distinct
        // array elements
        Set<Integer> B = new HashSet<Integer>();

        // Traverse the array, A[]
        for (int i = 0; i < N; i++)
        {
            B.add(A[i]);
        }

        ArrayList<Integer> b = new ArrayList<Integer>(B);

        // Count elements into the set
        int M = b.size();
        int i = 0;

        // If M is greater than 3
        if (M > 3)
        {    System.out.print("-1");}


        // If M is equal to 3
        else if (M == 3)
        {

            // Stores the first
            // smallest element
            int B_1 = b.get(i++);

            // Stores the second
            // smallest element
            int B_2 =  b.get(i++);

            // Stores the largest element
            int B_3 = b.get(i++);

            // IF difference between B_2 and B_1
            // is equal to B_3 and B_2
            if (B_2 - B_1 == B_3 - B_2)
            {
                System.out.print(B_2 - B_1);
            }
            else
            {
                System.out.print("-1");
            }


        }

        // If M is equal to 2
        else if (M == 2)
        {

            // Stores the smallest element
            int B_1 = b.get(i++);

            // Stores the largest element
            int B_2 = b.get(i++);

            // If difference is an even
            if ((B_2 - B_1) % 2 == 0)
            {
                System.out.print((B_2 - B_1) / 2);
            }

            else
            {
                System.out.print(B_2 - B_1);
            }
        }
        // If M is equal to 1
        else
        {
            System.out.print(0);
        }
    }

  // Driver code
    public static void main (String[] args)
    {

        // Given array
        int A[] = { 1, 3, 5, 1 };

        // Given size
        int N = A.length;


        // Print the required smallest integer
        findMinKToMakeAllEqual(N, A);
    }
}

// This code is contributed by rag2127

Python3

# Python3 program for the above approach

# Function to find smallest integer K such
# that incrementing or decrementing each
# element by K at most once makes all
# elements equal
def findMinKToMakeAllEqual(N, A):

    # Store distinct
    # array elements
    B = {}

    # Traverse the array, A[]
    for i in range(N):
        B[A[i]] = 1

    # Count elements into the set
    M = len(B)

    # Iterator to store first
    # element of B
    itr, i = list(B.keys()), 0

    # If M is greater than 3
    if (M > 3):
        print("-1")

    # If M is equal to 3
    elif (M == 3):

        # Stores the first
        # smallest element
        B_1, i = itr[i], i + 1

        # Stores the second
        # smallest element
        B_2, i = itr[i], i + 1

        # Stores the largest element
        B_3, i = itr[i], i + 1

        # IF difference between B_2 and B_1
        # is equal to B_3 and B_2
        if (B_2 - B_1 == B_3 - B_2):
            print(B_2 - B_1)
        else:
            print("-1")

    # If M is equal to 2
    elif (M == 2):

        # Stores the smallest element
        B_1, i = itr[i], i + 1

        # Stores the largest element
        B_2, i = itr[i], i + 1

        # If difference is an even
        if ((B_2 - B_1) % 2 == 0):
            print((B_2 - B_1) // 2)
        else:
            print(B_2 - B_1)

    # If M is equal to 1
    else:
        print(0)

# Driver Code
if __name__ == '__main__':

    # Given array
    A = [ 1, 3, 5, 1 ]

    # Given size
    N = len(A)

    # Print the required smallest integer
    findMinKToMakeAllEqual(N, A)

# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {

  // Function to find smallest integer K such that
  // incrementing or decrementing each element by
  // K at most once makes all elements equal
  static void findMinKToMakeAllEqual(int N, int[] A)
  {

    // Store distinct
    // array elements
    var B = new HashSet<int>();

    // Traverse the array, A[]
    for (int i = 0; i < N; i++) {
      B.Add(A[i]);
    }

    List<int> b = new List<int>(B);

    // Count elements into the set
    int M = b.Count;
    int j = 0;

    // If M is greater than 3
    if (M > 3) {
      Console.Write("-1");
    }

    // If M is equal to 3
    else if (M == 3) {

      // Stores the first
      // smallest element
      int B_1 = b[j++];

      // Stores the second
      // smallest element
      int B_2 = b[j++];

      // Stores the largest element
      int B_3 = b[j++];

      // IF difference between B_2 and B_1
      // is equal to B_3 and B_2
      if (B_2 - B_1 == B_3 - B_2) {
        Console.Write(B_2 - B_1);
      }
      else {
        Console.Write("-1");
      }
    }

    // If M is equal to 2
    else if (M == 2) {

      // Stores the smallest element
      int B_1 = b[j++];

      // Stores the largest element
      int B_2 = b[j++];

      // If difference is an even
      if ((B_2 - B_1) % 2 == 0) {
        Console.Write((B_2 - B_1) / 2);
      }

      else {
        Console.Write(B_2 - B_1);
      }
    }
    // If M is equal to 1
    else {
      Console.Write(0);
    }
  }

  // Driver code
  public static void Main(string[] args)
  {

    // Given array
    int[] A = { 1, 3, 5, 1 };

    // Given size
    int N = A.Length;

    // Print the required smallest integer
    findMinKToMakeAllEqual(N, A);
  }
}

// This code is contributed by chitranayal.

Javascript

<script>

// Javascript program for the above approach

// Function to find smallest integer K such that
// incrementing or decrementing each element by
// K at most once makes all elements equal
function findMinKToMakeAllEqual(N, A)
{

    // Store distinct
    // array elements
    var B = new Set();

    // Traverse the array, A[]
    for(var i = 0; i < N; i++)
        B.add(A[i]);

    // Count elements into the set
    var M = B.size;

    // Iterator to store first
    // element of B
    var itr = [...B].sort((a, b) => a - b);

    // If M is greater than 3
    if (M > 3)
        document.write("-1");

    // If M is equal to 3
    else if (M == 3)
    {

        // Stores the first
        // smallest element
        var B_1 = itr[0];

        // Stores the second
        // smallest element
        var B_2 = itr[1];

        // Stores the largest element
        var B_3 = itr[2];

        // IF difference between B_2 and B_1
        // is equal to B_3 and B_2
        if (B_2 - B_1 == B_3 - B_2)
            document.write(B_2 - B_1);
        else
            document.write("-1");
    }

    // If M is equal to 2
    else if (M == 2)
    {

        // Stores the smallest element
        var B_1 = itr[0];

        // Stores the largest element
        var B_2 =itr[1];

        // If difference is an even
        if ((B_2 - B_1) % 2 == 0)
            document.write(parseInt((B_2 - B_1) / 2));
        else
            document.write(B_2 - B_1);
    }

    // If M is equal to 1
    else
        document.write(0);
}

// Driver Code

// Given array
var A = [ 1, 3, 5, 1 ];

// Given size
var N = A.length;

// Print the required smallest integer
findMinKToMakeAllEqual(N, A);

// This code is contributed by noob2000

</script>

Output

2

Time Complexity: O(N * log(N))
Auxiliary Space: O(N)

Another Approach:

1. Find the minimum and maximum elements in the array.

• Initialize minElement and maxElement as the first element of the array.
• Iterate through the array and update minElement and maxElement accordingly.
• If the current element is less than minElement, update minElement.
• If the current element is greater than maxElement, update maxElement.

     2. Check if all elements are already equal.

• If minElement is equal to maxElement, then all elements are already equal, and the smallest positive integer K is 0. Return 0.

     3. Check if it’s possible to make all elements equal.

• If the difference between maxElement and minElement is odd, it’s not possible to make all elements equal by incrementing or decrementing them. Return -1.

     4. Calculate the value of ‘diff’ as half of the difference between maxElement and minElement.

• diff = (maxElement – minElement) / 2

     5. Check if it’s possible to make all elements equal by adding or subtracting ‘diff’.

• Iterate through the array.
• If the current element is not equal to minElement, maxElement, or minElement + diff, it’s not possible to make all elements equal. Return -1.

     6. Return ‘diff’ as the smallest positive integer K.

• ‘diff’ is the value by which each array element needs to be incremented or decremented to make all elements equal.

Below is the implementation of the above approach:

C++

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

int findSmallestK(vector<int>& arr) {
    int n = arr.size();

    // Find the minimum and maximum elements in the array
    int minElement = *min_element(arr.begin(), arr.end());
    int maxElement = *max_element(arr.begin(), arr.end());

    // Check if all elements are already equal
    if (minElement == maxElement) {
        return 0;
    }

    // Check if it's possible to make all elements equal
    if ((maxElement - minElement) % 2 != 0) {
        return -1;
    }

    int diff = (maxElement - minElement) / 2;

    // Check if it's possible to make all elements equal by adding or subtracting 'diff'
    for (int i = 0; i < n; i++) {
        if (arr[i] != minElement && arr[i] != maxElement && arr[i] != minElement + diff) {
            return -1;
        }
    }

    return diff;
}

int main() {
    vector<int> arr = { 5, 7, 9 };

    int smallestK = findSmallestK(arr);
    cout << smallestK << endl;

    return 0;
}

Java

import java.util.*;

public class GFG {

    // Function to find the smallest possible value of 'k'
    // such that all elements in the array can be made equal
    // by adding or subtracting 'k'
    public static int findSmallestK(List<Integer> arr) {
        int n = arr.size();

        // Find the minimum and maximum elements in the array
        int minElement = Collections.min(arr);
        int maxElement = Collections.max(arr);

        // Check if all elements are already equal
        if (minElement == maxElement) {
            return 0;
        }

        // Check if it's possible to make all elements equal
        if ((maxElement - minElement) % 2 != 0) {
            return -1;
        }

        int diff = (maxElement - minElement) / 2;

        // Check if it's possible to make all elements equal
        // by adding or subtracting 'diff'
        for (int i = 0; i < n; i++) {
            if (arr.get(i) != minElement && arr.get(i) != maxElement && arr.get(i) != minElement + diff) {
                return -1;
            }
        }

        return diff;
    }

    public static void main(String[] args) {
        List<Integer> arr = new ArrayList<>(Arrays.asList(5, 7, 9));

        int smallestK = findSmallestK(arr);
        System.out.println(smallestK);
    }
}

Python3

def find_smallest_k(arr):
    n = len(arr)

    # Find the minimum and maximum elements in the array
    min_element = min(arr)
    max_element = max(arr)

    # Check if all elements are already equal
    if min_element == max_element:
        return 0

    # Check if it's possible to make all elements equal
    if (max_element - min_element) % 2 != 0:
        return -1

    diff = (max_element - min_element) // 2

    # Check if it's possible to make all elements equal by adding or subtracting 'diff'
    for i in range(n):
        if arr[i] != min_element and arr[i] != max_element and arr[i] != min_element + diff:
            return -1

    return diff

if __name__ == "__main__":
    arr = [5, 7, 9]

    smallest_k = find_smallest_k(arr)
    print(smallest_k)

C#

using System;
using System.Collections.Generic;
using System.Linq;

namespace SmallestKEqualization
{
    class Program
    {
        static int FindSmallestK(List<int> arr)
        {
            int n = arr.Count;

            // Find the minimum and maximum elements in the list
            int minElement = arr.Min();
            int maxElement = arr.Max();

            // Check if all elements are already equal
            if (minElement == maxElement)
            {
                return 0;
            }

            // Check if it's possible to make all elements equal
            if ((maxElement - minElement) % 2 != 0)
            {
                return -1;
            }

            int diff = (maxElement - minElement) / 2;

            // Check if it's possible to make all elements equal by adding or subtracting 'diff'
            foreach (int num in arr)
            {
                if (num != minElement && num != maxElement && num != minElement + diff)
                {
                    return -1;
                }
            }

            return diff;
        }

        static void Main(string[] args)
        {
            List<int> arr = new List<int> { 5, 7, 9 };

            int smallestK = FindSmallestK(arr);
            Console.WriteLine(smallestK);

            // Pause execution to see the result
            Console.ReadKey();
        }
    }
}

Javascript

function findSmallestK(arr) {
    const n = arr.length;

    // Find the minimum and maximum elements in the array
    const minElement = Math.min(...arr);
    const maxElement = Math.max(...arr);

    // Check if all elements are already equal
    if (minElement === maxElement) {
        return 0;
    }

    // Check if it's possible to make all elements equal
    if ((maxElement - minElement) % 2 !== 0) {
        return -1;
    }

    const diff = (maxElement - minElement) / 2;

    // Check if it's possible to make all elements equal by adding or subtracting 'diff'
    for (let i = 0; i < n; i++) {
        if (arr[i] !== minElement && arr[i] !== maxElement && arr[i] !== minElement + diff) {
            return -1;
        }
    }

    return diff;
}

const arr = [5, 7, 9];

const smallestK = findSmallestK(arr);
console.log(smallestK);

Output

2

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