Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing

Question

I had an interesting job interview experience a while back. The question started really easy:

Q1: We have a bag containing numbers

Answer 1

I'd take a different approach to that question and probe the interviewer for more details about the larger problem he's trying to solve. Depending on the problem and the requirements surrounding it, the obvious set-based solution might be the right thing and the generate-a-list-and-pick-through-it-afterward approach might not.

For example, it might be that the interviewer is going to dispatch n messages and needs to know the k that didn't result in a reply and needs to know it in as little wall clock time as possible after the n-kth reply arrives. Let's also say that the message channel's nature is such that even running at full bore, there's enough time to do some processing between messages without having any impact on how long it takes to produce the end result after the last reply arrives. That time can be put to use inserting some identifying facet of each sent message into a set and deleting it as each corresponding reply arrives. Once the last reply has arrived, the only thing to be done is to remove its identifier from the set, which in typical implementations takes O(log k+1). After that, the set contains the list of k missing elements and there's no additional processing to be done.

This certainly isn't the fastest approach for batch processing pre-generated bags of numbers because the whole thing runs O((log 1 + log 2 + ... + log n) + (log n + log n-1 + ... + log k)). But it does work for any value of k (even if it's not known ahead of time) and in the example above it was applied in a way that minimizes the most critical interval.