Difference Between Ones and Zeros in Row and Column

Difference Between Ones and Zeros in Row and Column

What is this problem about?

The Difference Between Ones and Zeros in Row and Column coding problem tasks you with transforming an $m \times n$ binary matrix into a new "difference matrix." For each cell $(i, j)$ in the output matrix, the value is calculated as: onesRow_i + onesCol_j - zerosRow_i - zerosCol_j. Here, onesRow_i is the number of 1s in the $i$-th row, and zerosRow_i is the number of 0s in that row (similar definitions apply for columns).

Why is this asked in interviews?

Companies like Apple and Amazon use this to test basic Matrix interview patterns and simulation efficiency. While the formula seems complex, the problem tests if you can identify redundant calculations. A naive approach would count 1s and 0s for every single cell, leading to $O(m \times n \times (m+n))$ complexity. A more optimized approach uses precomputation to achieve $O(m \times n)$.

Algorithmic pattern used

The pattern used here is Precomputation (Simulation). You perform one pass over the matrix to pre-calculate the counts of 1s for every row and every column. Since the total number of elements in a row or column is fixed, zerosCount can be derived as total - onesCount. After precomputing these four arrays (or just two), you perform a second pass to populate the difference matrix using the provided formula in $O(1)$ per cell.

Example explanation

Suppose grid is:

0 1 1
1 0 1

1. Row counts (ones): Row 0 has two 1s, Row 1 has two 1s. onesRow = [2, 2].
2. Col counts (ones): Col 0 has one 1, Col 1 has one 1, Col 2 has two 1s. onesCol = [1, 1, 2].
3. Calculating cell (0,0):
   • onesRow[0] = 2, zerosRow[0] = 3 - 2 = 1.
   • onesCol[0] = 1, zerosCol[0] = 2 - 1 = 1.
   • Diff = $2 + 1 - 1 - 1 = 1$.
4. Repeat for all cells to fill the $2 \times 3$ result.

Common mistakes candidates make

• Recalculating every time: Scanning the whole row and column for each of the $m \times n$ cells.
• Memory Overhead: Creating too many intermediate matrices when a few arrays are sufficient.
• Off-by-one errors: Getting the dimensions $m$ and $n$ swapped during column or row summation.

Interview preparation tip

In matrix problems, always ask: "Can I pre-calculate row and column properties?" Most matrix transformation problems can be reduced from $O(N^3)$ to $O(N^2)$ by using linear arrays to cache sums, counts, or flags for each row and column.