See the original problem on HackerRank.
Given an unsorted array of \( n1 \) elements, it is missing exactly one element from the sequence of \( 1 \) to \( n\). Find the missing element.
Input Format
An integer \( n \) followed by \( n1 \) space separated numeric values.
Constraints
\( 1<n<10^6 \)
Output Format
The missing value.
Solutions
This is an example of problems that we call manifold at Coding Gym, since they give us room to find several alternative solutions.
Sort + Search
We sort the input array and we linearly scan the sequence to find the missing value. For example: In C++:


In Haskell:


Pros:
 easy to extend to a more generic domain (e.g. if we have a baseline array, we can easily adapt this solution to locate the element missing in the second array)
Cons:
 we pay sort  \( O(N \cdot logN) \)
Notes:
 we change the original array (what if not permitted?)
Variant: use binary search to locate the missing value, since the array is sorted.
Additional Storage
We could keep track of elements into a support storage (e.g. a flag array). This is just an example:


Pros:
 it's linear
 we do not modify the input array
 easy to extend to a more generic domain and also to duplicates (e.g. turn the vector into a frequency array)
Cons:
 we pay additional storage
Mathematics
We can exploit the problem domain to the limit.
 calculate the sum of all the numbers from \( 1 \) to \( N \). Call it \( expected \)
 calculate the sum of input numbers. Call if \( sum \)
 the missing value is just \( expected  sum \)
It's interesting to note that the sum from \( 1 \) to \( N \) is the Gauss Formula:
\( \dfrac{n(n+1)}{2} \)
In C++:


Pros:
 efficient
 no extra space
 we do not modify the input array
 the same idea (summation, not Gauss) can be reused to solve the same problem in more generic domains
Cons:
 what if \( \dfrac{n(n+1)}{2} \) overflows?
In Haskell there is no overflow, because the type Integer
is unbounded (also
known as bignum).


XOR
The following solution is very similar to the previous one but it never overflows.
It's based on XOR property:
 \( A \oplus B = C \)
 \( A \oplus C = B \)
 \( B \oplus C = A \)
That's the idea:
 suppose \( A \) is the result of XORing all the input elements
 suppose \( B \) is the missing value
 the result of XORing \( A \oplus B \) is \( C \)
 then \( C \) is the result of XORing all the elements from 1 to N
 we can easily find \( B \) as \( A \oplus C \)
Talk is cheap, let's look at some code:


In Haskell, but reducing with xor on a single list: the concatenation of the
original list plus numbers from 1 to n
.


Pros:
 never overflows
 the same idea can be reused to solve the same problem in more generic domains
Cons:
 tricky ;)