Maximum number of elements that can be removed such that MEX of the given array remains unchanged

Given an array arr[] of size N, the task is to count the maximum number of elements that can be removed from the given array without changing the MEX of the original array.

The MEX is the smallest positive integer that is not present in the array.

Examples:

Input: arr[] = {2, 3, 5, 1, 6}  
Output: 2 
Explanation:
The smallest positive integer which is not present in the array is 4.
Hence, MEX of the given array is 4.
Therefore, 5 and 6 can be removed without changing the MEX of the array.

Input: arr[] = {12, 4, 6, 1, 7, 2}  
Output: 4
Explanation:
The smallest positive integer which is not present in the array is 3.
Hence, MEX of the given array is 3.
Therefore, 4, 6, 7 and 12 can be removed without changing the MEX of the array. 

Naive Approach: The simplest approach is to sort the array and then traverse the array from i = 0 while arr[i] equals to i + 1. After that, print the answer as (N – i) which is the maximum number of elements that can be removed from the given array without changing its MEX.

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

Efficient Approach: To optimize the above approach, the idea is to use Hashing. Observe that, the maximum number of elements that can be removed is the number of elements that are greater than the MEX. Follow the steps below to solve the problem:

1. Initialize an array hash[] of length N+1 where hash[i] will be 1 if element i is present in the given array otherwise hash[i] = 0.
2. Initialize a variable mex with N + 1 to store MEX of the given array.
3. Traverse array hash[] over the range [1, N], and if for any index hash[i] equals to 0, update mex as mex = i and break out of the loop.
4. Print N – (mex – 1) as the maximum number of elements that can be removed from the given array.

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 maximum number
// of elements that can be removed
void countRemovableElem(
    int arr[], int N)
{
    // Initialize hash[] with 0s
    int hash[N + 1] = { 0 };

    // Initialize MEX
    int mex = N + 1;

    // Set hash[i] = 1, if i is
    // present in arr[]
    for (int i = 0; i < N; i++) {
        if (arr[i] <= N)
            hash[arr[i]] = 1;
    }

    // Find MEX from the hash
    for (int i = 1; i <= N; i++) {
        if (hash[i] == 0) {
            mex = i;
            break;
        }
    }

    // Print the maximum numbers
    // that can be removed
    cout << N - (mex - 1);
}

// Driver Code
int main()
{
    // Given array
    int arr[] = { 2, 3, 5, 1, 6 };

    // Size of the array
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    countRemovableElem(arr, N);

    return 0;
}

Java

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

class GFG{

// Function to find the maximum number
// of elements that can be removed
static void countRemovableElem(int[] arr, int N)
{

    // Initialize hash[] with 0s
    int[] hash = new int[N + 1];
    Arrays.fill(hash, 0);

    // Initialize MEX
    int mex = N + 1;

    // Set hash[i] = 1, if i is
    // present in arr[]
    for(int i = 0; i < N; i++)
    {
        if (arr[i] <= N)
            hash[arr[i]] = 1;
    }

    // Find MEX from the hash
    for(int i = 1; i <= N; i++)
    {
        if (hash[i] == 0)
        {
            mex = i;
            break;
        }
    }

    // Print the maximum numbers
    // that can be removed
    System.out.println(N - (mex - 1));
}

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

    // Given array
    int[] arr = { 2, 3, 5, 1, 6 };

    // Size of the array
    int N = arr.length;

    // Function Call
    countRemovableElem(arr, N);
}
}

// This code is contributed by akhilsaini

Python3

# Python3 program for the above approach

# Function to find the maximum number
# of elements that can be removed
def countRemovableElem(arr, N):

    # Initialize hash[] with 0s
    hash = [0] * (N + 1)

    # Initialize MEX
    mex = N + 1

    # Set hash[i] = 1, if i is
    # present in arr[]
    for i in range(0, N):
        if (arr[i] <= N):
            hash[arr[i]] = 1

    # Find MEX from the hash
    for i in range(1, N + 1):
        if (hash[i] == 0):
            mex = i
            break

    # Print the maximum numbers
    # that can be removed
    print(N - (mex - 1))

# Driver Code
if __name__ == '__main__':

    # Given array
    arr = [ 2, 3, 5, 1, 6 ]

    # Size of the array
    N = len(arr)

    # Function Call
    countRemovableElem(arr, N)

# This code is contributed by akhilsaini

C#

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

class GFG{

// Function to find the maximum number
// of elements that can be removed
static void countRemovableElem(int[] arr, int N)
{

    // Initialize hash[] with 0s
    int[] hash = new int[N + 1];
    Array.Fill(hash, 0);

    // Initialize MEX
    int mex = N + 1;

    // Set hash[i] = 1, if i is
    // present in arr[]
    for(int i = 0; i < N; i++)
    {
        if (arr[i] <= N)
            hash[arr[i]] = 1;
    }

    // Find MEX from the hash
    for(int i = 1; i <= N; i++)
    {
        if (hash[i] == 0)
        {
            mex = i;
            break;
        }
    }

    // Print the maximum numbers
    // that can be removed
    Console.WriteLine(N - (mex - 1));
}

// Driver Code
public static void Main()
{

    // Given array
    int[] arr = { 2, 3, 5, 1, 6 };

    // Size of the array
    int N = arr.Length;

    // Function Call
    countRemovableElem(arr, N);
}
}

// This code is contributed by akhilsaini

Javascript

<script>

// Javascript program to implement
// the above approach

// Function to find the maximum number
// of elements that can be removed
function countRemovableElem(arr, N)
{

    // Initialize hash[] with 0s
    let hash = [];
    for(let i = 0; i < N; i++)
    {
        hash[i] = 0;
    }

    // Initialize MEX
    let mex = N + 1;

    // Set hash[i] = 1, if i is
    // present in arr[]
    for(let i = 0; i < N; i++)
    {
        if (arr[i] <= N)
            hash[arr[i]] = 1;
    }

    // Find MEX from the hash
    for(let i = 1; i <= N; i++)
    {
        if (hash[i] == 0)
        {
            mex = i;
            break;
        }
    }

    // Print the maximum numbers
    // that can be removed
    document.write(N - (mex - 1));
}
    // Driver Code

        // Given array
    let arr = [2, 3, 5, 1, 6 ];

    // Size of the array
    let N = arr.length;

    // Function Call
    countRemovableElem(arr, N);

</script>

Output:

2

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