# Identifying the missing value

Topics/tags: The joy of code, technical, interview questions, short

Today another student asked me to give a mock interview [1]. This interview involved smaller questions, rather than one big problem. One of the questions I asked falls in the category of questions that try to test intuition, but mostly test whether you’ve seen the problem before or not. Nonetheless, I thought the conversation we had was fun, and so I’m reporting it here.

You have an array of size n-1 containing values between 1 and n. No value appears twice. One value does not appear at all. Identify the missing value.

If your brain works just right, you’ve come up with an efficient solution almost immediately.

I’ll give you a bit of time [3] to think about it.

One.

Two.

Three.

Four.

Five.

Six.

Seven.

Eight.

Nine.

Ten.

Eleven.

That should be enough. Do you have a solution? Here’s a typical one.

Sort the array from smallest to largest. If the first value is not one, one is missing. If the last value is not n, n is missing. Otherwise, iterate the array until you find a gap.

How efficient is the algorithm? One should, of course, give both time and space complexity. The answer is It depends. On what? On the sorting algorithm you use. If you use one of the efficient in-place compare-and-swap algorithms [4], the algorithm requires O(n log n) time and O(1) space. However, for numbers in a restricted range, bucket sort or radix sort may be more appropriate. In that case, the algorithm requires O(n) time and O(n) space.

As is my habit, I asked Can you do better? [6]. In this case, it’s pretty clear that you have to look at every element at least once, so better means O(n) time and O(1) space.

Can you do better?

Eleven.

Ten.

Nine.

Eight.

Seven.

Six.

Five.

Four.

Three.

Two.

One.

Eventually, the student came up with an in-place bucket-sort algorithm and an argument that it ran in O(n).

But I’m mean. So I asked him to assume that the array was immutable. Do you have a solution in that case? Let’s pause again.

Cosine.

Tangent.

Secant.

Sine.

Three.

Point.

One.

Four.

One.

Five.

Nine.

I believe the answer most interviewers want their subjects to come up with is as follows.

I know that the sum of the numbers from 1 to n is n*(n+1)/2. So I can sum the numbers in the array and subtract it from that number, giving the missing number.

Wasn’t that fun?

Postscript: If I give away all of my mock interview questions in these musings, what will I do when I have to do my next mock interview? Perhaps I’ll need to build up my stock.

[1] No, I did not mock them all that much, except when (a) they forgot about heap sort and (b) they claimed that lookup in hash tables is linear [2].

[2] I prefer to hear the adjectives amortized and expected when talking about hash tables.

[3] Or at least a bit of space.

[4] Heap sort and Quicksort are the traditional choices [5].

[5] Yes, this is the point that the student forgot about heap sort.

[6] Can I do better? is a question I encourage students to ask themselves regularly when they write algorithms.

Version 1.0 of 2018-12-19.