Find Missing and Repeated Values

Find Missing and Repeated Values

What is this problem about?

The Find Missing and Repeated Values interview question is a grid-based identification task. You are given an $n imes n$ matrix containing integers from $1$ to $n^2$. Because of a mistake, one number appears twice (the repeated value) and one number is missing. Your goal is to identify both numbers. This Find Missing and Repeated Values coding problem is a fundamental exercise in frequency counting and mathematical identification.

Why is this asked in interviews?

Companies like Microsoft and Amazon ask this to test basic data structure usage and mathematical reasoning. It evaluates if you can solve a problem using either a Hash Table interview pattern (for $O(N^2)$ space) or a Math interview pattern (for $O(1)$ space). It's a great introductory problem to assess a candidate's ability to optimize for space while maintaining linear time complexity.

Algorithmic pattern used

This problem can be solved using Frequency Counting or Math (Sum and Square Sum).

1. Frequency Counting: Use a hash map or a boolean array of size $n^2+1$. Iterate through the grid, marking seen numbers. The one seen twice is the repeated value; the one never seen is the missing value.
2. Mathematical Approach:
   • Calculate the actual sum ($S$) and the expected sum of $1 \dots n^2$.
   • Calculate the actual square sum ($SS$) and the expected square sum.
   • Use the two resulting equations $(x-y)$ and $(x^2-y^2)$ to solve for the repeated ($x$) and missing ($y$) values.

Example explanation

Grid $2 imes 2$: [[1, 3], [2, 2]] (Expected: 1, 2, 3, 4)

• Frequencies: 1: once, 2: twice, 3: once, 4: zero times.
• Repeated: 2.
• Missing: 4. Result: [2, 4].

Common mistakes candidates make

• Space Complexity: Using a full 2D matrix for tracking instead of a simple 1D frequency array.
• Integer Overflow: Forgetting that the sum of squares for large $n$ (e.g., $n=50$) can exceed the limits of a standard 32-bit integer.
• Nested Loops: While necessary to read the grid ($O(n^2)$), avoid nested loops during the search phase if you can find the answer in a single pass.

Interview preparation tip

Whenever you need to find "missing" or "duplicate" numbers in a fixed range, always consider the Math approach (sums and XORs). It demonstrates a high level of optimization awareness that interviewers value.