Minimum Sensors to Cover Grid

Minimum Sensors to Cover Grid

What is this problem about?

The Minimum Sensors to Cover Grid problem asks you to determine the minimum number of sensors needed to cover every row in a grid, where each sensor placed at a certain column position covers a fixed range of rows. The exact coverage rule follows a mathematical pattern based on the sensor's position relative to grid dimensions. This Minimum Sensors to Cover Grid coding problem is fundamentally a mathematics and coverage reasoning challenge.

Why is this asked in interviews?

Cisco, Microsoft, and Amazon ask this to evaluate pure mathematical reasoning and the ability to derive formulas without code scaffolding. It tests whether candidates can abstract a coverage problem into a simple arithmetic formula rather than simulating placement. The math interview pattern applies cleanly, and correctness depends on understanding the specific coverage definition.

Algorithmic pattern used

The key insight is that sensors placed on specific rows or columns provide symmetric or asymmetric coverage depending on grid parity. Derive the formula by working out which rows each sensor position covers, then determine the minimum number of non-overlapping placements needed to cover all rows. For most variants: sensors cover ranges determined by row count, and the minimum count follows a ceiling division formula.

Example explanation

Suppose a grid of n=7 rows. A sensor placed at position p covers rows [p-2, p+2] (5 rows). To cover all 7 rows with minimum sensors:

• Sensor at row 2: covers rows 0-4.
• Sensor at row 6: covers rows 4-7 (clipped to 0-6). Total: 2 sensors.

Formula generalizes as ceil(n / coverage_per_sensor), adjusted for edge effects.

Common mistakes candidates make

• Simulating sensor placements manually instead of deriving the formula.
• Not accounting for boundary effects (sensors near grid edges have reduced effective range).
• Off-by-one in coverage range calculation.
• Overcomplicating with a greedy or DP approach when direct math suffices.

Interview preparation tip

Math interview problems reward candidates who can think formulaically before coding. When you see "minimum number of [units] to cover [range]," immediately think ceiling division. Derive the coverage per unit, compute total needed, and verify with edge cases (very small grids, single-sensor coverage). Clean mathematical reasoning — showing your work clearly on a whiteboard — is often more impressive than working code for this type of problem.