Find Unique Binary String

Find Unique Binary String

1. What is this problem about?

The Find Unique Binary String interview question asks you to find a binary string of length $n$ that does not appear in a given list of $n$ unique binary strings. Since there are $2^n$ possible binary strings and you are only given $n$ of them, a unique string is guaranteed to exist. The goal is to find any such string efficiently.

2. Why is this asked in interviews?

Companies like Microsoft and Google ask the Find Unique Binary String coding problem to see if a candidate is familiar with Cantor's Diagonal Argument. While the problem can be solved with Backtracking or a Hash Set, the diagonal approach is an $O(N)$ mathematical "hack" that demonstrates high-level problem-solving efficiency and theoretical knowledge.

3. Algorithmic pattern used

This problem can be solved using Cantor's Diagonalization or Backtracking.

1. Cantor's Diagonal Argument ($O(N)$):
   • Construct a result string where the $i^{th}$ character is the opposite of the $i^{th}$ character of the $i^{th}$ string in the input list.
   • This guarantees the result string is different from the $1^{st}$ string (at index 0), the $2^{nd}$ string (at index 1), and so on.
2. Backtracking ($O(2^N)$): Systematically generate binary strings and check their existence in a set until a unique one is found.

4. Example explanation

Input: ["01", "10"] (n=2)

• Diagonal Logic:
  • $i=0$: nums[0] is "01". Take the $0^{th}$ char ('0') and flip it to '1'.
  • $i=1$: nums[1] is "10". Take the $1^{st}$ char ('0') and flip it to '1'.
• Result: "11". Check: "11" is not in ["01", "10"]. Correct!

5. Common mistakes candidates make

• Inefficiency: Using a full $O(2^N)$ search when $n$ could be large enough to make it slow (though usually $n$ is small for this specific problem).
• Set Overhead: Building a hash set of all $2^n$ possibilities, which uses $O(2^n)$ memory.
• Lexicographical search: Trying to convert strings to integers and finding the first missing number, which requires sorting ($O(N \log N)$).

6. Interview preparation tip

Cantor's Diagonalization is a "trick" that shows up in several "unique sequence" or "unseen item" problems. Memorizing this pattern can help you provide an $O(N)$ solution that will highly impress interviewers looking for optimal bit-level logic.