Find the Missing Number

Given an array arr[] of size n-1 with distinct integers in the range of [1, n]. This array represents a permutation of the integers from 1 to n with one element missing. Find the missing element in the array.

Examples:

Input: arr[] = [8, 2, 4, 5, 3, 7, 1]
Output: 6
Explanation: All the numbers from 1 to 8 are present except 6.

Input: arr[] = [1, 2, 3, 5]
Output: 4
Explanation: Here the size of the array is 4, so the range will be [1, 5]. The missing number between 1 to 5 is 4

[Naive Approach] Linear Search for Missing Number - O(n^2) Time and O(1) Space

This approach iterates through each number from 1 to n (where n is the size of the array + 1) and checks if the number is present in the array. For each number, it uses a nested loop to search the array. If a number is not found, it is returned as the missing number.

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[Better Approach] Using Hashing - O(n) Time and O(n) Space

This approach uses a hash array (or frequency array) to track the presence of each number from 1 to n in the input array. It first initializes a hash array to store the frequency of each element. Then, it iterates through the hash array to find the number that is missing (i.e., the one with a frequency of 0).

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[Expected Approach 1] Using Sum of n terms Formula - O(n) Time and O(1) Space

The sum of the first n natural numbers is given by the formula (n * (n + 1)) / 2. The idea is to compute this sum and subtract the sum of all elements in the array from it to get the missing number.

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[Expected Approach 2] Using XOR Operation - O(n) Time and O(1) Space

XOR of a number with itself is 0 i.e. x ^ x = 0 and the given array arr[] has numbers in range [1, n]. This means that the result of XOR of first n natural numbers with the XOR of all the array elements will be the missing number. To do so, calculate XOR of first n natural numbers and XOR of all the array arr[] elements, and then our result will be the XOR of both the resultant values.

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