Find the Minimum Area to Cover All Ones II

Find the Minimum Area to Cover All Ones II

What is this problem about?

In the Find the Minimum Area to Cover All Ones II coding problem, you are asked to divide the grid into exactly three non-overlapping rectangles such that every '1' in the grid is covered by at least one of these rectangles. You need to minimize the sum of the areas of these three rectangles.

Why is this asked in interviews?

This "Hard" problem from Microsoft and Salesforce tests your ability to handle complex Enumeration and spatial partitioning. Unlike the first version, you must now consider all possible ways to split a 2D space into three regions. It evaluation your mastery of recursion or multi-step logic to solve optimization problems with geometric constraints.

Algorithmic pattern used

This problem is solved by Enumerating Split Points. There are only a few ways to split a rectangle into three non-overlapping sub-rectangles:

1. Two horizontal cuts (forming three horizontal strips).
2. Two vertical cuts (forming three vertical strips).
3. One horizontal cut, then one of the resulting parts split by a vertical cut (forming a 'T' shape or its rotations).
4. One vertical cut, then one part split by a horizontal cut. For each possible cut configuration, you use the logic from "Minimum Area I" to find the bounding box of 1s within each sub-region and sum their areas.

Example explanation

Consider a large grid. A vertical cut at $x=10$ splits it into $[0, 10]$ and $[11, Max]$. Then, a horizontal cut at $y=5$ splits the second part into $[11, Max] imes [0, 5]$ and $[11, Max] imes [6, Max]$. You calculate the minimum area covering all 1s in these three disjoint regions and repeat for all possible $x$ and $y$.

Common mistakes candidates make

• Missing configurations: Only considering strips and forgetting the 'T' or 'rotated-T' split shapes.
• Redundant Calculations: Re-calculating the minimum area for the same sub-region multiple times. Caching the results of "Min Area I" for various grid slices can help.
• Complexity: Attempting to use DP for arbitrary $K$ rectangles when $K=3$ allows for exhaustive enumeration of split patterns.

Interview preparation tip

For spatial split problems, draw out the possible cut patterns. For $K=3$, there are exactly 6 fundamental layouts. Once you identify these, the problem reduces to correctly applying the 1-rectangle solution to various sub-windows of the grid.