Special Positions in a Binary Matrix

Special Positions in a Binary Matrix

What is this problem about?

The Special Positions in a Binary Matrix interview question asks you to count the number of positions $(r, c)$ in a binary matrix such that matrix[r][c] == 1 and all other elements in the same row $r$ and the same column $c$ are 0. It is a problem of identifying unique "1"s that stand alone in their respective cross-sections of the matrix.

Why is this asked in interviews?

This Special Positions in a Binary Matrix coding problem is a great test of a candidate's ability to handle 2D data structures efficiently. Companies like Goldman Sachs and Amazon use it to evaluate if a candidate can avoid redundant work. A naive approach might check every '1' by scanning its entire row and column, but a more experienced developer will use pre-computation to solve it in linear time relative to the number of elements.

Algorithmic pattern used

The primary pattern is Pre-computation or Array frequency counting. Instead of scanning the row and column for every '1', you can perform two passes:

1. One pass to count the number of 1s in each row and store these counts in an array.
2. Another pass to count the number of 1s in each column and store them in another array.
3. A final pass through the matrix: if matrix[r][c] == 1, rowCount[r] == 1, and colCount[c] == 1, then the position is "special."

Example explanation

Matrix:

1 0 0
0 0 1
1 0 0

• Row counts: Row0=1, Row1=1, Row2=1
• Column counts: Col0=2, Col1=0, Col2=1

Checking each '1':

• (0,0): Row count is 1, but Column count is 2. Not special.
• (1,2): Row count is 1, Column count is 1. Special!
• (2,0): Row count is 1, Column count is 2. Not special. Result: 1 special position.

Common mistakes candidates make

• Brute Force Efficiency: Scanning the row and column for every '1' found, leading to $O(M \times N \times (M+N))$ complexity instead of $O(M \times N)$.
• Space Optimization: Forgetting that they only need two small arrays (one for rows, one for columns) to store the counts.
• Incorrect Counting: Accidentally counting the element itself twice or mis-indexing the row/column arrays.

Interview preparation tip

For matrix problems involving row/column properties, always consider if you can "project" the properties onto 1D arrays. Storing the sum or count of each row/column is a very common technique that reduces the complexity of many grid-based problems.