Minimize maximum difference between any pair of array elements by exactly one removal

Given an array arr[] consisting of N integers(N > 2), the task is to minimize the maximum difference between any pair of elements (arr[i], arr[j]) by removing exactly one element.

Examples:

Input: arr[] = {1, 3, 4, 7}
Output: 3
Explanation:
Removing the element 7 from array, modifies the array  to {1, 3, 4}.
Here (4, 1) has difference = 4 – 1 = 3 which is minimum possible maximum difference. 

Input: arr[] = {1, 2, 3, 4}
Output: 2

Naive Approach: The simplest approach to solve the given problem is to remove every element one by one and check which element gives the minimized maximum difference between every pair of elements.

Time Complexity: O(N³)
Auxiliary Space: O(N) 

Efficient Approach: The above approach can also be optimized by observing the fact that the maximum difference is equal to the difference between the maximum and the minimum element of the given array. So, removing the maximum element or removing the minimum element always gives the optimal answer.

Therefore, the idea is to sort the given array in ascending order and print the minimum of the value (arr[N – 2] – arr[0]) and (arr[N – 1] – arr[1]) as the resultant minimized maximum difference.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum maximum
// difference after removing one element
int findMinDifference(int arr[], int n)
{
    // Stores the resultant minimized
    // maximum difference
    int ans = 0;

    // Sort the given array
    sort(arr, arr + n);

    // Find the minimum maximum
    // difference
    ans = min(arr[n - 2] - arr[0],
              arr[n - 1] - arr[1]);

    // Return the result
    return ans;
}

// Driver Code
int main()
{
    int arr[] = { 1, 3, 4, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);
    cout << findMinDifference(arr, N);

    return 0;
}

Java

// Java program for the above approach

import java.io.*;
import java.util.*;

class GFG {

  // Function to find the minimum maximum
  // difference after removing one element
  static int findMinDifference(int []arr, int n)
  {

    // Stores the resultant minimized
    // maximum difference
    int ans = 0;

    // Sort the given array
    Arrays.sort(arr);

    // Find the minimum maximum
    // difference
    ans = Math.min(arr[n - 2] - arr[0],
                   arr[n - 1] - arr[1]);

    // Return the result
    return ans;
  }

  // Driver Code

  public static void main (String[] args) {

    int []arr = { 1, 3, 4, 7 };
    int N = arr.length;


    System.out.println(findMinDifference(arr, N));
  }

}

// This code is contributed by unknown2108

Python3

# Python3 program for the above approach

# Function to find the minimum maximum
# difference after removing one element
def findMinDifference(arr, n):

    # Stores the resultant minimized
    # maximum difference
    ans = 0

    # Sort the given array
    arr = sorted(arr)

    # Find the minimum maximum
    # difference
    ans = min(arr[n - 2] - arr[0],
              arr[n - 1] - arr[1])

    # Return the result
    return ans

# Driver Code
if __name__ == '__main__':

    arr = [ 1, 3, 4, 7 ]
    N = len(arr)

    print (findMinDifference(arr, N))

# 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 the minimum maximum
// difference after removing one element
static int findMinDifference(int []arr, int n)
{

    // Stores the resultant minimized
    // maximum difference
    int ans = 0;

    // Sort the given array
    Array.Sort(arr);

    // Find the minimum maximum
    // difference
    ans = Math.Min(arr[n - 2] - arr[0],
                   arr[n - 1] - arr[1]);

    // Return the result
    return ans;
}

// Driver Code
public static void Main()
{
    int []arr = { 1, 3, 4, 7 };
    int N = arr.Length;

    Console.Write(findMinDifference(arr, N));
}
}

// This code is contributed by ipg2016107

Javascript

<script>
// Javascript program for the above approach

// Function to find the minimum maximum
// difference after removing one element
function findMinDifference(arr, n)
{

    // Stores the resultant minimized
    // maximum difference
    let ans = 0;

    // Sort the given array
    arr.sort(function(a,b){return a-b;});

    // Find the minimum maximum
    // difference
    ans = Math.min(arr[n - 2] - arr[0],
                   arr[n - 1] - arr[1]);

    // Return the result
    return ans;
}

// Driver Code
let arr=[1, 3, 4, 7];
let N = arr.length;
document.write(findMinDifference(arr, N));

// This code is contributed by patel2127
</script>

Output

3

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

Efficient Approach:- 

• As in the above approach we are either removing the maximum element or the minimum element
• So we can do this work without sorting 
• We have to just find out the 2 maximum and 2 minimum elements from the array
• Answer will be minimum of (2nd largest-first minimum) or (first lasgest-second minimum).

Implementation:-

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum maximum
// difference after removing one element
int findMinDifference(int arr[], int n)
{

      //taking variables to store 2 max and 2 min elements
      int firstmin=INT_MAX,secondmin=INT_MAX,firstmax=INT_MIN,secondmax=INT_MIN;

      //iterating over array
      for(int i=0;i<n;i++)
    {

          //if found min element than firstmin
          if(arr[i]<firstmin)
        {
              //updating secondmin
              secondmin=firstmin;
              //updating firstmin
              firstmin=arr[i];
        }

          //if found element small than secondmin
          else if(arr[i]<secondmin)
        {
              secondmin=arr[i];
        }

          //if found element larger than firstmax
          if(arr[i]>firstmax)
        {
              //updating second max
              secondmax=firstmax;
              //updating first max
              firstmax=arr[i];
        }

          //if found element greater than second max
          else if(arr[i]>secondmax)
          {
              secondmax=arr[i];
        }
    }

      //returning answer
      return min(secondmax-firstmin,firstmax-secondmin);

}

// Driver Code
int main()
{
    int arr[] = { 1, 3, 4, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);
    cout << findMinDifference(arr, N);

    return 0;
}
//This code is contributed by shubhamrajput6156

Java

// Java program for the above approach

import java.lang.*;
import java.util.*;

class Main {
    // Function to find the minimum maximum
    // difference after removing one element
    static int findMinDifference(int arr[], int n)
    {
            // taking variables to store 2 max and 2 min
            // elements
            int firstmin
            = Integer.MAX_VALUE,
            secondmin = Integer.MAX_VALUE,
            firstmax = Integer.MIN_VALUE,
            secondmax = Integer.MIN_VALUE;

        // iterating over array
        for (int i = 0; i < n; i++) {

            // if found min element than firstmin
            if (arr[i] < firstmin) {
                // updating secondmin
                secondmin = firstmin;
                // updating firstmin
                firstmin = arr[i];
            }

            // if found element small than secondmin
            else if (arr[i] < secondmin) {
                secondmin = arr[i];
            }

            // if found element larger than firstmax
            if (arr[i] > firstmax) {
                // updating second max
                secondmax = firstmax;
                // updating first max
                firstmax = arr[i];
            }

            // if found element greater than second max
            else if (arr[i] > secondmax) {
                secondmax = arr[i];
            }
        }

        // returning answer
        return Math.min(secondmax - firstmin,
                        firstmax - secondmin);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 1, 3, 4, 7 };
        int N = arr.length;
        System.out.println(findMinDifference(arr, N));
    }
}

Python3

# Python program for the above approach
import sys

# Function to find the minimum maximum
# difference after removing one element
def findMinDifference(arr, n):

    # taking variables to store 2 max and 2 min elements
    firstmin = sys.maxsize
    secondmin = sys.maxsize
    firstmax = -sys.maxsize - 1
    secondmax = -sys.maxsize - 1

    for i in range(n):

        # if found min element than firstmin
        if arr[i] < firstmin:
            # updating secondmin
            secondmin = firstmin
            #updating firstmin
            firstmin = arr[i]

        # if found element small than secondmin
        elif arr[i] < secondmin:
            secondmin = arr[i]

        # if found element larger than firstmax
        if arr[i] > firstmax:
            # update second max
            secondmax = firstmax
            # update first max
            firstmax = arr[i]

        # if found element greater than second max
        elif arr[i] > secondmax:
            secondmax = arr[i]

    # returning answer
    return min(secondmax - firstmin, firstmax - secondmin)

# driver code
arr = [1, 3, 4, 7]
N = len(arr)
print(findMinDifference(arr, N))

# This code is contributed by redmoonz.

Javascript

// Java program for the above approach

    // Function to find the minimum maximum
    // difference after removing one element
    const findMinDifference = (arr, n) => {

    // taking variables to store 2 max and 2 min
    // elements
    let firstmin = Number.MAX_SAFE_INTEGER;
    let secondmin = Number.MAX_SAFE_INTEGER;
    let firstmax = Number.MIN_SAFE_INTEGER;
    let secondmax = Number.MIN_SAFE_INTEGER;

// iterating over array
    for (let i = 0; i < n; i++) {

    // if found min element than firstmin
        if (arr[i] < firstmin) {
            secondmin = firstmin;
            firstmin = arr[i];
        } else if (arr[i] < secondmin) {
            secondmin = arr[i];
        }

        if (arr[i] > firstmax) {
            secondmax = firstmax;
            firstmax = arr[i];

            // if found element greater than second max
        } else if (arr[i] > secondmax) {
            secondmax = arr[i];
        }
    }

    // returning answer
    return Math.min(secondmax - firstmin, firstmax - secondmin);
}

const arr = [1, 3, 4, 7];
const n = arr.length;
console.log(findMinDifference(arr, n));

C#

// C# program for the above approach

using System;

public class GFG{

    // Function to find the minimum maximum
    // difference after removing one element
    public static int findMinDifference(int[] arr, int n)
    {

          //taking variables to store 2 max and 2 min elements
          int firstmin = int.MaxValue;
          int secondmin = int.MaxValue;
          int firstmax = int.MinValue;
          int secondmax = int.MinValue;

          //iterating over array
          for (int i = 0;i < n;i++)
          {

              //if found min element than firstmin
              if (arr[i] < firstmin)
              {
                  //updating secondmin
                  secondmin = firstmin;
                  //updating firstmin
                  firstmin = arr[i];
              }

              //if found element small than secondmin
              else if (arr[i] < secondmin)
              {
                  secondmin = arr[i];
              }

              //if found element larger than firstmax
              if (arr[i] > firstmax)
              {
                  //updating second max
                  secondmax = firstmax;
                  //updating first max
                  firstmax = arr[i];
              }

              //if found element greater than second max
              else if (arr[i] > secondmax)
              {
                  secondmax = arr[i];
              }
          }

          //returning answer
          return Math.Min(secondmax - firstmin,firstmax - secondmin);

    }

    // Driver Code
    internal static void Main()
    {
        int[] arr = {1, 3, 4, 7};

        int N = arr.Length;
        Console.Write(findMinDifference(arr, N));

    }
    //This code is contributed by bhardwajji
}

Output

3

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