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3648. Minimum Sensors to Cover Grid

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You are given n × m grid and an integer k.

A sensor placed on cell (r, c) covers all cells whose Chebyshev distance from (r, c) is at most k.

The Chebyshev distance between two cells (r1, c1) and (r2, c2) is max(|r1 − r2|,|c1 − c2|).

Your task is to return the minimum number of sensors required to cover every cell of the grid.

 

Example 1:

Input: n = 5, m = 5, k = 1

Output: 4

Explanation:

Placing sensors at positions (0, 3), (1, 0), (3, 3), and (4, 1) ensures every cell in the grid is covered. Thus, the answer is 4.

Example 2:

Input: n = 2, m = 2, k = 2

Output: 1

Explanation:

With k = 2, a single sensor can cover the entire 2 * 2 grid regardless of its position. Thus, the answer is 1.

 

Constraints:

  • 1 <= n <= 103
  • 1 <= m <= 103
  • 0 <= k <= 103
Math

Hints

Hint 1
Let s = 2*k + 1, the side length each sensor covers.
Hint 2
The minimum number is ceil(n / s) * ceil(m / s).