Claudio Cherubino's blog Life of a Googler

The missing number

There is an interesting series of programming job interview challenges proposed by Dev102.com, which is now at its tenth puzzle:

This week question is pretty easy. Your input is an unsorted list of n numbers ranging from 1 to n+1, all of the numbers are unique, meaning that a number can’t appear twice in that list. According to those rules, one of the numbers is missing and you are asked to provide the most efficient method to find that missing number. Notice that you can’t allocate another helper list, the amount of memory that you are allowed to allocate is O(1). Don’t forget to mention the complexity of your algorithm…

I'm not sure I understood correctly the constraint related to the memory allocation.

In my opinion, when they say we are limited to O(1), they mean that we can only allocate a single numeric variable and not any other data structure.

According to this interpretation, the solution is quite easy.

First of all, we take our only variable and store into it the sum of all the numbers between 1 and n + 1, which can be easily computed remembering that 1 + 2 + ... + n = n (n + 1) / 2.

Then, we subtract each element of the array from this value, eventually the result is actually our missing number.

The functional implementation of this imperative algorithm is straightforward:

#light

let sum n = ((n + 1) * (n + 2)) / 2

let answer nums = (nums |> List.length |> sum) - (List.reduce_left (+) nums)

Let's talk about the complexity of the algorithm.

If the length of the input list list is known, evaluating the sum of the first n natural numbers takes O(1), otherwise we have to scan the entire list, i.e. O(n).

Then we have to subtract each element of the list, and this takes another iteration, so another O(n).

Therefore we have O(n) + O(n), which is definitely O(n), and can be easily optimized a little by scanning the list only once.

The only problem left is: "am I following the instructions"?