Find a number where it appears exactly N/2 times

Question

Here is one of my interview question. Given an array of N elements and where an element appears exactly N/2 times and the rest N/2 elements are unique

Answer 1

Peter is exactly right. Here is a more formal way of restating his proof:

Let set S be a set containing N elements. It is the union of two sets: p, which contains a symbol α repeated N/2 times, and q, which contains N/2 unique symbols ω₁..ω_n/2. S = p ∪ q.

Assume there is an algorithm that can detect your duplicated number in log(n) comparisons in the worst case for all N > 2. In the worst case means that there does not exist any subset r ⊂ S such that |r| = log₂ N where α ∉ r.

However because S = p ∪ q, there are |p| many elements ≠ α in S. |p| = N/2, so ∀ N/2 such that N/2 ≥ log₂N, there must exist at least one set r ⊂ S such that |r| = log₂N and α ∉ r. This is the case for any N ≥ 3. This contradicts the assumption above, so there cannot be any such algorithm.

QED.