Find the missing number in an array - Strivers A2Z DSA Course

Problem Statement: Given an integer N and an array of size N-1 containing N-1 numbers between 1 to N. Find the number(between 1 to N), that is not present in the given array.

Examples:

Example 1:
Input Format: N = 5, array[] = {1,2,4,5}
Result: 3
Explanation: In the given array, number 3 is missing. So, 3 is the answer.

Example 2:
Input Format: N = 3, array[] = {1,3}
Result: 2
Explanation: In the given array, number 2 is missing. So, 2 is the answer.

Solution

Disclaimer: Don’t jump directly to the solution, try it out yourself first.
Problem Link 1.
Problem Link 2.

Naive Approach(Brute-force approach):

Intuition: For each number between 1 to N, we will try to find it in the given array using linear search. And if we don’t find any of them, we will return the number.

Approach:

The algorithm steps are as follows:

We will run a loop(say i) from 1 to N.
For each integer, i, we will try to find it in the given array using linear search.
1. For that, we will run another loop to iterate over the array and consider a flag variable to indicate if the element exists in the array. Flag = 1 means the element is present and flag = 0 means the element is missing.
2. Initially, the flag value will be set to 0. While iterating the array, if we find the element, we will set the flag to 1 and break out from the loop.
  Now, for any number i, if we cannot find it, the flag will remain 0 even after iterating the whole array and we will return the number.

Dry run:

Assume given array = {1, 3} and N = 3.

Note: For each number, before starting the linear search, we set the flag to 0.

For i = 1, flag = 0
We will try to find 1 in the array using linear search and in the first index, we can find it and the flag will be 1. So, it is not the missing number.

For i = 2, flag = 0
We will try to find 2 in the given array using linear search. But number 2 is not present and the flag will remain 0. So, we will return 2 as it is the missing number.

Code:

C++ Code

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

int missingNumber(vector<int>&a, int N) {

    // Outer loop that runs from 1 to N:
    for (int i = 1; i <= N; i++) {

        // flag variable to check
        //if an element exists
        int flag = 0;

        //Search the element using linear search:
        for (int j = 0; j < N - 1; j++) {
            if (a[j] == i) {

                // i is present in the array:
                flag = 1;
                break;
            }
        }

        // check if the element is missing
        //i.e flag == 0:

        if (flag == 0) return i;
    }

    // The following line will never execute.
    // It is just to avoid warnings.
    return -1;
}

int main()
{
    int N = 5;
    vector<int> a = {1, 2, 4, 5};
    int ans = missingNumber(a, N);
    cout << "The missing number is: " << ans << endl;
    return 0;
}

Output: The missing number is: 3

Time Complexity: O(N²), where N = size of the array+1.
Reason: In the worst case i.e. if the missing number is N itself, the outer loop will run for N times, and for every single number the inner loop will also run for approximately N times. So, the total time complexity will be O(N²).

Space Complexity: O(1) as we are not using any extra space.

Java Code

import java.util.*;

public class tUf {
    public static int missingNumber(int []a, int N) {

        // Outer loop that runs from 1 to N:
        for (int i = 1; i <= N; i++) {

            // flag variable to check
            //if an element exists
            int flag = 0;

            //Search the element using linear search:
            for (int j = 0; j < N - 1; j++) {
                if (a[j] == i) {

                    // i is present in the array:
                    flag = 1;
                    break;
                }
            }

            // check if the element is missing
            //i.e flag == 0:

            if (flag == 0) return i;
        }

        // The following line will never execute.
        // It is just to avoid warnings.
        return -1;
    }

    public static void main(String args[]) {
        int N = 5;
        int a[] = {1, 2, 4, 5};

        int ans = missingNumber(a, N);
        System.out.println("The missing number is: " + ans);
    }

}

Output: The missing number is: 3

Space Complexity: O(1) as we are not using any extra space.

Better Approach (using Hashing):

Intuition: Using the hashing technique, we will store the frequency of each element of the given array. Now, the number( i.e. between 1 to N) for which the frequency will be 0, will be returned. Refer to the article link to know more about hashing.

Approach:

The algorithm steps are as follows:

The range of the number is 1 to N. So, we need a hash array of size N+1 (as we want to store the frequency of N as well).
Now, for each element in the given array, we will store the frequency in the hash array.
After that, for each number between 1 to N, we will check the frequencies. And for any number, if the frequency is 0, we will return it.

Dry run:

Assume the given array = {1,3} and N = 3. The hash array will look like the following:

We can clearly see that for index 2 the frequency is 0 and so 2 is our answer.

Code:

C++ Code

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

int missingNumber(vector<int>&a, int N) {

    int hash[N + 1] = {0}; //hash array

    // storing the frequencies:
    for (int i = 0; i < N - 1; i++)
        hash[a[i]]++;

    //checking the freqencies for numbers 1 to N:
    for (int i = 1; i <= N; i++) {
        if (hash[i] == 0) {
            return i;
        }
    }

    // The following line will never execute.
    // It is just to avoid warnings.
    return -1;
}

int main()
{
    int N = 5;
    vector<int> a = {1, 2, 4, 5};
    int ans = missingNumber(a, N);
    cout << "The missing number is: " << ans << endl;
    return 0;
}

Output: The missing number is: 3

Time Complexity: O(N) + O(N) ~ O(2*N), where N = size of the array+1.
Reason: For storing the frequencies in the hash array, the program takes O(N) time complexity and for checking the frequencies in the second step again O(N) is required. So, the total time complexity is O(N) + O(N).

Space Complexity: O(N), where N = size of the array+1. Here we are using an extra hash array of size N+1.

Java Code

import java.util.*;

public class tUf {
    public static int missingNumber(int []a, int N) {

        int hash[] = new int[N + 1]; //hash array

        // storing the frequencies:
        for (int i = 0; i < N - 1; i++)
            hash[a[i]]++;

        //checking the freqencies for numbers 1 to N:
        for (int i = 1; i <= N; i++) {
            if (hash[i] == 0) {
                return i;
            }
        }

        // The following line will never execute.
        // It is just to avoid warnings.
        return -1;
    }

    public static void main(String args[]) {
        int N = 5;
        int a[] = {1, 2, 4, 5};

        int ans = missingNumber(a, N);
        System.out.println("The missing number is: " + ans);
    }

}

Output: The missing number is: 3

Space Complexity: O(N), where N = size of the array+1. Here we are using an extra hash array of size N+1.

Optimal Approach:

There are two optimal approaches for this particular question. They are the following:
1. Summation approach
2. XOR(^) approach

Let’s discuss each of them.

1. Summation Approach:

Intuition: We know that the summation of the first N numbers is (N*(N+1))/2. We can say this S1. Now, in the given array, every number between 1 to N except one number is present. So, if we add the numbers of the array (say S2), the difference between S1 and S2 will be the missing number. Because, while adding all the numbers of the array, we did not add that particular number that is missing.

Sum of first N numbers(S1) = (N*(N+1))/2
Sum of all array elements(S2) = i = 0n-2a[i]
The missing number = S1-S2

Approach:

The steps are as follows:

We will first calculate the summation of first N natural numbers(i.e. 1 to N) using the specified formula.
Then we will add all the array elements using a loop.
Finally, we will consider the difference between the summation of the first N natural numbers and the sum of the array elements.

Dry run:

Assume the given array is: {1, 2, 4, 5} and N = 5.
Summation of 1 to 5 = (5*(5+1))/2 = 15
Summation of array elements = 12
So, the difference will be = 15 - 12 = 3. 
And the missing number is also 3.

Code:

C++ Code

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

int missingNumber(vector<int>&a, int N) {

    //Summation of first N numbers:
    int sum = (N * (N + 1)) / 2;

    //Summation of all array elements:
    int s2 = 0;
    for (int i = 0; i < N - 1; i++) {
        s2 += a[i];
    }

    int missingNum = sum - s2;
    return missingNum;
}

int main()
{
    int N = 5;
    vector<int> a = {1, 2, 4, 5};
    int ans = missingNumber(a, N);
    cout << "The missing number is: " << ans << endl;
    return 0;
}

Output: The missing number is: 3

Time Complexity: O(N), where N = size of array+1.
Reason: Here, we need only 1 loop to get the sum of the array elements. The loop runs for approx. N times. So, the time complexity is O(N).

Space Complexity: O(1) as we are not using any extra space.

Java Code

import java.util.*;

public class tUf {
    public static int missingNumber(int []a, int N) {

        //Summation of first N numbers:
        int sum = (N * (N + 1)) / 2;

        //Summation of all array elements:
        int s2 = 0;
        for (int i = 0; i < N - 1; i++) {
            s2 += a[i];
        }

        int missingNum = sum - s2;
        return missingNum;
    }

    public static void main(String args[]) {
        int N = 5;
        int a[] = {1, 2, 4, 5};

        int ans = missingNumber(a, N);
        System.out.println("The missing number is: " + ans);
    }
}

Output: The missing number is: 3

Time Complexity: O(N), where N = size of array+1.
Reason: Here, we need only 1 loop to get the sum of the array elements. The loop runs for approx. N times. So, the time complexity is O(N).

Space Complexity: O(1) as we are not using any extra space.

2. XOR Approach:

Intuition: Two important properties of XOR are the following:

XOR of two same numbers is always 0 i.e. a ^ a = 0. ←Property 1.
XOR of a number with 0 will result in the number itself i.e. 0 ^ a = a. ←Property 2

Now, let’s XOR all the numbers between 1 to N.
xor1 = 1^2^…….^N

Let’s XOR all the numbers in the given array.
xor2 = 1^2^……^N (contains all the numbers except the missing one).

Now, if we again perform XOR between xor1 and xor2, we will get:
xor1 ^ xor2 = (1^1)^(2^2)^……..^(missing Number)^…..^(N^N)

Here all the numbers except the missing number will form a pair and each pair will result in 0 according to the XOR property. The result will be = 0 ^ (missing number) = missing number (according to property 2).

So, if we perform the XOR of the numbers 1 to N with the XOR of the array elements, we will be left with the missing number.

Approach:

The steps are as follows:

We will first run a loop(say i) from 0 to N-2(as the length of the array = N-1).
Inside the loop, xor2 variable will calculate the XOR of array elements
i.e. xor2 = xor2 ^ a[i].
And the xor1 variable will calculate the XOR of 1 to N-1 i.e. xor1 = xor1 ^ (i+1).
After the loop ends we will XOR xor1 and N to get the total XOR of 1 to N.
Finally, the answer will be the XOR of xor1 and xor2.

Dry run:

Assume the given array is: {1, 2, 4, 5} and N = 5.
XOR of (1 to 5) i.e. xor1 = (1^2^3^4^5)
XOR of array elements i.e. xor2 = (1^2^4^5)
XOR of xor1 and xor2 = (1^2^3^4^5) ^ (1^2^4^5)
			= (1^1)^(2^2)^(3)^(4^4)^(5^5)
			= 0^0^3^0^0 = 0^3 = 3.
The missing number is 3.

Code:

C++ Code

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

int missingNumber(vector<int>&a, int N) {

    int xor1 = 0, xor2 = 0;

    for (int i = 0; i < N - 1; i++) {
        xor2 = xor2 ^ a[i]; // XOR of array elements
        xor1 = xor1 ^ (i + 1); //XOR up to [1...N-1]
    }
    xor1 = xor1 ^ N; //XOR up to [1...N]

    return (xor1 ^ xor2); // the missing number
}

int main()
{
    int N = 5;
    vector<int> a = {1, 2, 4, 5};
    int ans = missingNumber(a, N);
    cout << "The missing number is: " << ans << endl;
    return 0;
}

Output: The missing number is: 3

Time Complexity: O(N), where N = size of array+1.
Reason: Here, we need only 1 loop to calculate the XOR. The loop runs for approx. N times. So, the time complexity is O(N).

Space Complexity: O(1) as we are not using any extra space.

Java Code

import java.util.*;

public class tUf {
    public static int missingNumber(int []a, int N) {

        int xor1 = 0, xor2 = 0;

        for (int i = 0; i < N - 1; i++) {
            xor2 = xor2 ^ a[i]; // XOR of array elements
            xor1 = xor1 ^ (i + 1); //XOR up to [1...N-1]
        }
        xor1 = xor1 ^ N; //XOR up to [1...N]

        return (xor1 ^ xor2); // the missing number
    }

    public static void main(String args[]) {
        int N = 5;
        int a[] = {1, 2, 4, 5};

        int ans = missingNumber(a, N);
        System.out.println("The missing number is: " + ans);
    }
}

Output: The missing number is: 3

Time Complexity: O(N), where N = size of array+1.
Reason: Here, we need only 1 loop to calculate the XOR. The loop runs for approx. N times. So, the time complexity is O(N).

Space Complexity: O(1) as we are not using any extra space.

Note: Among the optimal approaches, the XOR approach is slightly better than the summation one because the term (N * (N+1))/2 used in the summation method cannot be stored in an integer if the value of N is big (like 10⁵). In that case, we have to use some bigger data types. But we will face no issues like this while using the XOR approach.

Special thanks to KRITIDIPTA GHOSH for contributing to this article on takeUforward. If you also wish to share your knowledge with the takeUforward fam, please check out this article. If you want to suggest any improvement/correction in this article please mail us at [email protected]