Minimum Sensors to Cover Grid - Solution & Explanation

Problem Overview: You are given an m x n grid and must place the minimum number of sensors so that every cell is covered. A single sensor covers a fixed square region around its position, which means one sensor protects multiple nearby cells. The task reduces to determining how many such coverage blocks are needed to cover the entire grid.

Approach 1: Grid Simulation / Greedy Placement (O(m * n) time, O(1) space)

A straightforward strategy scans the grid from top-left to bottom-right. Whenever you encounter a cell that has not been covered yet, place a sensor at the top-left position of the next coverage block and mark the cells inside that block as covered. This behaves like tiling the grid with fixed-size squares. Implementation typically uses nested loops and manually advances indices to skip already-covered regions. While the logic works and is easy to reason about, it still iterates through most cells of the grid, leading to O(m * n) time.

This approach demonstrates the core idea of coverage and is useful during interviews to show how you reason about grid traversal and greedy placement decisions.

Approach 2: Mathematical Tiling Insight (O(1) time, O(1) space)

Instead of simulating coverage, observe that each sensor effectively covers a 3 x 3 block of cells. The entire grid can therefore be treated as a tiling problem. Along the rows, you need enough sensors to cover m cells using blocks of size 3, which requires ceil(m / 3). Similarly, columns require ceil(n / 3). Since sensors cover both directions simultaneously, the total number of sensors becomes:

ceil(m / 3) * ceil(n / 3)

This works because any grid can be partitioned into these coverage blocks, and partial leftover rows or columns still require one full sensor placement. The result is a constant-time formula that avoids iterating through the grid entirely.

The key insight comes from recognizing the repeating coverage pattern and converting a grid simulation into a direct formula using math. Problems like this often appear under grid optimization or tiling patterns and can sometimes also be framed as simplified implementation tasks.

Recommended for interviews: Start with the simulation idea to show you understand how sensors cover the grid. Then derive the tiling observation that each sensor covers a fixed block and compute the count using ceiling division. Interviewers typically expect the mathematical solution because it reduces the problem from O(m * n) to O(1) while keeping space usage constant.