MEX (Minimum Excluded) in Competitive Programming

MEX of a sequence or an array is the smallest non-negative integer that is not present in the sequence.

Example:

• If we have a sequence S[] = {1,2,0,4,5} then MEX (S) = 3.
• If we have a sequence S[] ={2,0,3,10} then MEX (S) = 1.
• If we have a sequence S[] ={1,2,3,4,5} then MEX (S) = 0.

Note: The MEX of an array of size N cannot be greater than N since the MEX of an array is the smallest non-negative integer not present in the array and array having size N can only cover integers from 0 to N-1. Therefore, the MEX of an array with N elements cannot be greater than N itself.

How to Find MEX of an array (Without Updates) ?

Given an array arr of size N, the task is to determine the MEX of the array.

Approach:

The idea is to use a map to store frequency of each element of an array. Then, iterate through non-negative integers from 0 to N, and return the first integer with a frequency of 0 in the map. This integer represents the smallest non-negative integer not present in the array, which is the MEX.

Below is the implementation of above approach

C++

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

// Function to find the MEX of the array
int mex(vector<int>& arr, int N)
{
    // Create a map to store the frequency of each element
    map<int, int> mp;
    for (int i = 0; i < N; i++) {
        mp[arr[i]]++;
    }

    // Initialize MEX to 0
    int mex = 0;

    // Iterate through non-negative integers from 0 to N
    for (int i = 0; i <= N; i++) {
        // Find the first integer with a frequency of 0 in the map
        if (mp[i] == 0) {
            mex = i;
            break;
        }
    }

    // Return MEX as the answer
    return mex;
}

int main()
{
    // Example array
    vector<int> arr = { 1, 0, 2, 4 };
    int N = arr.size();

    // Call the mex function and print the result
    cout << "MEX of the array: " << mex(arr, N) << endl;
    return 0;
}

Java

import java.util.HashMap;

public class MinimumExcludedValue {

    // Function to find the MEX of the array
    static int mex(int[] arr, int N) {
        // Create a map to store the frequency of each element
        HashMap<Integer, Integer> mp = new HashMap<>();
        for (int i = 0; i < N; i++) {
            mp.put(arr[i], mp.getOrDefault(arr[i], 0) + 1);
        }

        // Initialize MEX to 0
        int mex = 0;

        // Iterate through non-negative integers from 0 to N
        for (int i = 0; i <= N; i++) {
            // Find the first integer with a frequency of 0 in the map
            if (!mp.containsKey(i)) {
                mex = i;
                break;
            }
        }

        // Return MEX as the answer
        return mex;
    }

    public static void main(String[] args) {
        // Example array
        int[] arr = {1, 0, 2, 4};
        int N = arr.length;

        // Call the mex function and print the result
        System.out.println("MEX of the array: " + mex(arr, N));
    }
}

Python

# Function to find the MEX of the array
def mex(arr):
    # Create a dictionary to store the frequency of each element
    freq_map = {}
    for num in arr:
        freq_map[num] = freq_map.get(num, 0) + 1

    # Initialize MEX to 0
    mex_val = 0

    # Iterate through non-negative integers from 0 to N
    N = len(arr)
    for i in range(N + 1):
        # Find the first integer with a frequency of 0 in the dictionary
        if i not in freq_map:
            mex_val = i
            break

    # Return MEX as the answer
    return mex_val


# Main function
if __name__ == "__main__":
    # Example array
    arr = [1, 0, 2, 4]

    # Call the mex function and print the result
    print("MEX of the array:", mex(arr))

C#

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

class MexAlgorithm
{
    static int Mex(int[] arr, int N)
    {
        Dictionary<int, int> freqMap = new Dictionary<int, int>();

        // Store the frequency of each element in the array
        for (int i = 0; i < N; i++)
        {
            if (freqMap.ContainsKey(arr[i]))
            {
                freqMap[arr[i]]++;
            }
            else
            {
                freqMap[arr[i]] = 1;
            }
        }

        int mex = 0;

        // Iterate through non-negative integers from 0 to N
        for (int i = 0; i <= N; i++)
        {
            // Find the first integer with a frequency of 0 in the dictionary
            if (!freqMap.ContainsKey(i) || freqMap[i] == 0)
            {
                mex = i;
                break;
            }
        }

        return mex;
    }

    static void Main()
    {
        // Example array
        int[] arr = { 1, 0, 2, 4 };
        int N = arr.Length;

        // Call the Mex function and print the result
        Console.WriteLine("MEX of the array: " + Mex(arr, N));
    }
}

Javascript

// Function to find the MEX of the array
function mex(arr, N) {
    // Create a map to store the frequency of each element
    const mp = new Map();
    for (let i = 0; i < N; i++) {
        if (mp.has(arr[i])) {
            mp.set(arr[i], mp.get(arr[i]) + 1);
        } else {
            mp.set(arr[i], 1);
        }
    }

    // Initialize MEX to 0
    let mex = 0;

    // Iterate through non-negative integers from 0 to N
    for (let i = 0; i <= N; i++) {
        // Find the first integer with a frequency of 0 in the map
        if (!mp.has(i) || mp.get(i) === 0) {
            mex = i;
            break;
        }
    }

    // Return MEX as the answer
    return mex;
}

// Main function
function main() {
    // Example array
    const arr = [1, 0, 2, 4];
    const N = arr.length;

    // Call the mex function and print the result
    console.log("MEX of the array:", mex(arr, N));
}

// Call the main function
main();

Output

MEX of the array: 3

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

How to Find MEX of an array (With Updates) ?

Given an array arr of size N, and q queries. In each query you are given two numbers (idx, val), you have to update the element at index idx with value val. The task is to determine the MEX of the array after each query.

Approach:

The idea is to use a map to store frequency of each element of an array. Then insert all the numbers in the set from 0 to N which are not present in the array. For each query, update the array and maintain the map and set accordingly. The MEX for each query is determined by selecting the smallest element from the set and added to the result.

Step-by-step approach:

• Create a map to store the frequency of array elements and a set to store the elements(from 0 to N) which are not present in the array.
• For each query.
  • Decrement the frequency of the element which is being replaced. If the frequency becomes 0, that element is added to set since it is not present in the array.
  • Increment the frequency of the updated element. Remove the updated element from the set if the set contains it, since the updated element will be present in the set.
• The MEX for each query is determined by selecting the smallest element from the set and added to the result.

Below is the implementation of above approach:

C++

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

// Function to calculate the MEX of an array after a series
// of queries
vector<int> mex(vector<int>& arr, int N,
                vector<vector<int> > queries, int q)
{
    // Initialize a map to store element frequencies
    map<int, int> mp;

    // Initialize a set to store non-present elements
    set<int> st;

    // Populate the map for the initial array
    for (int i = 0; i < N; i++) {
        mp[arr[i]]++;
    }

    // Add the elements to set which are not present in the
    // array.
    for (int i = 0; i <= N; i++) {
        if (mp[i] == 0)
            st.insert(i);
    }

    // Initialize a vector to store MEX values after each
    // query
    vector<int> mex;

    // Process each query
    for (int i = 0; i < q; i++) {
        int idx = queries[i][0];
        int val = queries[i][1];

        // Update the frequency of element which being
        // replaced.
        mp[arr[idx]]--;

        // If the frequency of element which is being
        // replaced becomes 0, add it to set.
        if (mp[arr[idx]] == 0) {
            st.insert(arr[idx]);
        }
        arr[idx] = val;
        mp[arr[idx]]++;

        // If the updated element is in the set, remove it
        if (st.find(arr[idx]) != st.end())
            st.erase(arr[idx]);

        // Calculate the MEX and add it to the result vector
        int ans = *st.begin();
        mex.push_back(ans);
    }

    return mex;
}

// Driver Code
int main()
{
    // Example array
    vector<int> arr = { 1, 0, 2, 4, 7 };
    int N = arr.size();

    // Example queries
    vector<vector<int> > queries
        = { { 1, 3 }, { 0, 0 }, { 4, 1 }, { 2, 10 } };
    int q = queries.size();

    // Call the 'mex' function to calculate MEX values after
    // each query
    vector<int> ans = mex(arr, N, queries, q);

    // Print the results
    for (int i = 0; i < q; i++) {
        cout << "MEX of the array after query " << i + 1
             << " : " << ans[i] << endl;
    }

    return 0;
}

Java

import java.util.*;

public class Main {

    // Function to calculate the MEX of an array after a series
    // of queries
    static List<Integer> mex(List<Integer> arr, int N,
                              List<List<Integer>> queries, int q) {
        // Initialize a map to store element frequencies
        Map<Integer, Integer> mp = new HashMap<>();

        // Initialize a set to store non-present elements
        TreeSet<Integer> st = new TreeSet<>();

        // Populate the map for the initial array
        for (int i = 0; i < N; i++) {
            mp.put(arr.get(i), mp.getOrDefault(arr.get(i), 0) + 1);
        }

        // Add the elements to set which are not present in the array
        for (int i = 0; i <= N; i++) {
            if (!mp.containsKey(i))
                st.add(i);
        }

        // Initialize a list to store MEX values after each query
        List<Integer> mex = new ArrayList<>();

        // Process each query
        for (int i = 0; i < q; i++) {
            int idx = queries.get(i).get(0);
            int val = queries.get(i).get(1);

            // Update the frequency of element which is being replaced
            mp.put(arr.get(idx), mp.get(arr.get(idx)) - 1);

            // If the frequency of the element which is being replaced becomes 0, add it to set
            if (mp.get(arr.get(idx)) == 0) {
                st.add(arr.get(idx));
            }
            arr.set(idx, val);
            mp.put(val, mp.getOrDefault(val, 0) + 1);

            // If the updated element is in the set, remove it
            if (st.contains(arr.get(idx)))
                st.remove(arr.get(idx));

            // Calculate the MEX and add it to the result list
            int ans = st.first();
            mex.add(ans);
        }

        return mex;
    }

    // Driver Code
    public static void main(String[] args) {
        // Example array
        List<Integer> arr = new ArrayList<>(Arrays.asList(1, 0, 2, 4, 7));
        int N = arr.size();

        // Example queries
        List<List<Integer>> queries = new ArrayList<>(
                Arrays.asList(
                        Arrays.asList(1, 3),
                        Arrays.asList(0, 0),
                        Arrays.asList(4, 1),
                        Arrays.asList(2, 10)
                )
        );
        int q = queries.size();

        // Call the 'mex' function to calculate MEX values after each query
        List<Integer> ans = mex(arr, N, queries, q);

        // Print the results
        for (int i = 0; i < q; i++) {
            System.out.println("MEX of the array after query " + (i + 1) + " : " + ans.get(i));
        }
    }
}

Python3

def mex(arr, N, queries, q):
    # Initialize a dictionary to store element frequencies
    mp = {}
    # Initialize a set to store non-present elements
    st = set()

    # Populate the dictionary for the initial array
    for num in arr:
        mp[num] = mp.get(num, 0) + 1

    # Add the elements to set which are not present in the array.
    for i in range(N + 1):
        if i not in mp:
            st.add(i)

    mex = []  # Initialize a list to store MEX values after each query

    # Process each query
    for i in range(q):
        idx, val = queries[i]

        # Update the frequency of the element being replaced.
        mp[arr[idx]] -= 1

        # If the frequency of the element being replaced becomes 0, add it to the set.
        if mp[arr[idx]] == 0:
            st.add(arr[idx])

        arr[idx] = val

        # Update the frequency of the new element.
        mp[val] = mp.get(val, 0) + 1

        # If the updated element is in the set, remove it.
        if arr[idx] in st:
            st.remove(arr[idx])

        # Calculate the MEX and add it to the result list
        ans = min(st)
        mex.append(ans)

    return mex

# Example array
arr = [1, 0, 2, 4, 7]
N = len(arr)

# Example queries
queries = [
    [1, 3],
    [0, 0],
    [4, 1],
    [2, 10]
]
q = len(queries)

# Call the 'mex' function to calculate MEX values after each query
ans = mex(arr, N, queries, q)

# Print the results
for i in range(q):
    print(f"MEX of the array after query {i + 1}: {ans[i]}")

C#

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

class Program
{
    // Function to calculate the MEX of an array after a series
    // of queries
    static List<int> Mex(List<int> arr, int N, List<List<int>> queries, int q)
    {
        // Initialize a dictionary to store element frequencies
        Dictionary<int, int> mp = new Dictionary<int, int>();

        // Initialize a HashSet to store non-present elements
        HashSet<int> st = new HashSet<int>();

        // Populate the dictionary for the initial array
        foreach (int num in arr)
        {
            if (mp.ContainsKey(num))
                mp[num]++;
            else
                mp[num] = 1;
        }

        // Add the elements to set which are not present in the array.
        for (int i = 0; i <= N; i++)
        {
            if (!mp.ContainsKey(i))
                st.Add(i);
        }

        // Initialize a list to store MEX values after each query
        List<int> mex = new List<int>();

        // Process each query
        foreach (List<int> query in queries)
        {
            int idx = query[0];
            int val = query[1];

            // Update the frequency of element which is being replaced.
            mp[arr[idx]]--;

            // If the frequency of element which is being replaced becomes 0, add it to set.
            if (mp[arr[idx]] == 0)
                st.Add(arr[idx]);

            arr[idx] = val;

            if (mp.ContainsKey(val))
                mp[val]++;
            else
                mp[val] = 1;

            // If the updated element is in the set, remove it
            if (st.Contains(val))
                st.Remove(val);

            // Calculate the MEX and add it to the result list
            int ans = st.Min();
            mex.Add(ans);
        }

        return mex;
    }

    // Driver Code
    static void Main(string[] args)
    {
        // Example array
        List<int> arr = new List<int> { 1, 0, 2, 4, 7 };
        int N = arr.Count;

        // Example queries
        List<List<int>> queries = new List<List<int>>
        {
            new List<int> { 1, 3 },
            new List<int> { 0, 0 },
            new List<int> { 4, 1 },
            new List<int> { 2, 10 }
        };
        int q = queries.Count;

        // Call the 'Mex' function to calculate MEX values after each query
        List<int> ans = Mex(arr, N, queries, q);

        // Print the results
        for (int i = 0; i < q; i++)
        {
            Console.WriteLine($"MEX of the array after query {i + 1} : {ans[i]}");
        }
    }
}

Javascript

function calculateMex(arr, N, queries, q) {
    // Initialize a dictionary to store element frequencies
    let mp = new Map();

    // Initialize a Set to store non-present elements
    let st = new Set();

    // Populate the dictionary for the initial array
    arr.forEach((num) => {
        mp.set(num, (mp.get(num) || 0) + 1);
    });

    // Add the elements to set which are not present in the array
    for (let i = 0; i <= N; i++) {
        if (!mp.has(i)) {
            st.add(i);
        }
    }

    // Initialize a list to store MEX values after each query
    let mex = [];

    // Process each query
    for (let i = 0; i < q; i++) {
        let idx = queries[i][0];
        let val = queries[i][1];

        // Update the frequency of element which is being replaced
        mp.set(arr[idx], mp.get(arr[idx]) - 1);

        // If the frequency of element which is being replaced becomes 0, add it to set
        if (mp.get(arr[idx]) === 0) {
            st.add(arr[idx]);
        }

        arr[idx] = val;

        if (mp.has(val)) {
            mp.set(val, mp.get(val) + 1);
        } else {
            mp.set(val, 1);
        }

        // If the updated element is in the set, remove it
        if (st.has(val)) {
            st.delete(val);
        }

        // Calculate the MEX and add it to the result list
        let ans = Math.min(...Array.from(st));
        mex.push(ans);
    }

    return mex;
}

// Example array
let arr = [1, 0, 2, 4, 7];
let N = arr.length;

// Example queries
let queries = [
    [1, 3],
    [0, 0],
    [4, 1],
    [2, 10]
];
let q = queries.length;

// Call the 'calculateMex' function to calculate MEX values after each query
let ans = calculateMex(arr, N, queries, q);

// Print the results
for (let i = 0; i < q; i++) {
    console.log(`MEX of the array after query ${i + 1} : ${ans[i]}`);
}

Output

MEX of the array after query 1 : 0
MEX of the array after query 2 : 1
MEX of the array after query 3 : 5
MEX of the array after query 4 : 2

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

How to solve MEX related problems in Competitive Programming ?

MEX Related tasks usually involves finding MEX of the array while updating its elements. Some of the common characteristics of MEX related tasks include:

• Array Updates – In general we are generally given queries in which we have either add, remove or update the values of array and the task involves finding the MEX after each such query.
• Data Structures- To optimally compute MEX after each query, set and maps are generally used because both provide fast lookup time for the elements (log n), also they keep elements sorted making it easier to find the minimum non-present element (MEX).

Let us consider few problems involving MEX:

Problem 1: Given an empty array arr[] and a integer m. You are given q queries. In i_th query you add a element y_i to the array. After each query you can perform the following operation any number of times: Choose an element arr_j and set arr_j to arr_j +m or arr_j -m. The task is maximize the value of MEX of array arr after each query.

The problem can be solved in following way:

Observation: We need to firstly observe that after applying the operation: arr_j +m or arr_j -m on an element arr_j, its modulo with m still remains the same after the operation.

Keep the track of Frequency: We keep track of frequency of remainders when divided by m, i.e., for each number [0,m-1] keep track of its frequency with each query. Let freq_{0 ,} freq₁ ,……, freq_m-1 be the frequency of remainders 0, 1, ….. m-1.

Query Updates: During each query, we update the frequency of the remainder, by keeping the set sorted based on frequency and remainder. For example, if the query adds an element y, we calculate y % m to determine its remainder when divided by m and then update its frequency.

MEX Calculation: In each query after updating the frequency of remainder we will find the minimum of {freq_{0 ,} freq₁ ,……, freq_m-1 }. If frq_j is remainder with minimum frequency the maximum MEX will be equal to: j*frq_j + j.

For example:- Suppose m=5 and frequency of remainders 0, 1, 2, 3, 4 are 3, 4, 4, 2 ,5 respectively. The minimum frequency is of remainder 3 with frequency 2. Since frequency of remainder 3 is 2, elements 3 and 8 will be present in the array by performing the operations. So the MEX will be 13 (2*10+3) in this case.

Problem 2: Given an array of n integers each integer lies in range [0,N]. The task is to make the array exactly {0, 1, 2,… , N-1} in almost 2*n operations. In one operation you can pick any element of the array and replace it with MEX of the array.

The problem can be solved in following way:

Observation: MEX of the array after each operation will lie between [0,N] and in the final array for each index i, arr_i = i. Let cnt be the number of indexes such that arr_i is not equal to i. Thus the cnt represents number of indexes for which we have to change the value of that index to its correct value.

Perform the Operations: In each operation we will firstly find the MEX of the array. Now we have 2 cases:
Case 1: When Mex of the array is between [0,N-1]: In this case we will simply put the MEX of the array in its correct index, i.e., if min_exc represents the MEX of the array then we will put min_exc in the index min_exc, this will decrease the value of cnt by 1.
Case 2: When Mex of the array is N: In this case, we will put the MEX of the array in any of the index for which arr_i is not equal to i. The value of cnt will remain the same.

Correctness: The above approach will convert array arr to {0,1,2, …. , N-1} in almost 2*N operations. The maximum value of cnt can be N, and the above approach decreases the value of cnt will by 1 in almost 2 operations because each time we encounter Case 2(MEX of the array is N), next time we perform the operation, we will get Case 1(MEX of the array is not N) since N was inserted in the array in last operation so MEX will not be N. Thus, we will Case 1 in which value of cnt will be decreased by 1. Hence, it takes almost 2*N operations to make the value of cnt 0.

Practice Problems on MEX:

Minimum operations to make all Array elements 0 by MEX replacement
Minimum operations to make the MEX of the given set equal to x
Find the Prefix-MEX Array for given Array
Rearrange array elements to maximize the sum of MEX of all prefix arrays
Maximum number of elements that can be removed such that MEX of the given array remains unchanged
Maximum MEX from all subarrays of length K
Minimum operations for same MEX
Maximize MEX by adding or subtracting K from Array elements
MEX of generated sequence of N+1 integers where ith integer is XOR of (i-1) and K
Maximize sum of MEX values of each node in an N-ary Tree
Find MEX of every subtree in given Tree
Minimum size of the array with MEX as A and XOR of the array elements as B
Rearrange Array to maximize sum of MEX of all Subarrays starting from first index