Minimum Cost for Cutting Cake I

Minimum Cost for Cutting Cake I

What is this problem about?

The Minimum Cost for Cutting Cake I problem involves an $m \times n$ cake that needs to be cut into $1 \times 1$ individual squares. Each horizontal and vertical line has an associated cost for cutting. When you make a cut, it splits the current piece(s) of cake, and any future cuts on those pieces will cost more because you have to perform the cut on each separate piece. The goal is to find the minimum total cost to reduce the cake to individual squares.

Why is this asked in interviews?

This is a standard Minimum Cost for Cutting Cake interview question at Google. It tests your ability to apply a Greedy strategy to a problem that might initially look like it requires complex Dynamic Programming. It evaluates if you can recognize that the most expensive cuts should be made as early as possible to minimize their repetition.

Algorithmic pattern used

The Greedy interview pattern with Sorting is the key.

1. Sort both horizontal and vertical costs in descending order.
2. Maintain two counters: h_pieces = 1 and v_pieces = 1.
3. Always pick the largest available cost (whether horizontal or vertical).
4. If it's a horizontal cut, the total cost increases by cost * v_pieces, and you increment h_pieces.
5. If it's a vertical cut, the total cost increases by cost * h_pieces, and you increment v_pieces.

Example explanation

Cake $3 \times 3$. Horizontal costs: [2, 3], Vertical costs: [4, 5].

1. Sort all costs: V:5, V:4, H:3, H:2.
2. First cut (V:5): Cost = 5 * 1 (h_pieces) = 5. v_pieces = 2.
3. Second cut (V:4): Cost = 4 * 1 (h_pieces) = 4. v_pieces = 3.
4. Third cut (H:3): Cost = 3 * 3 (v_pieces) = 9. h_pieces = 2.
5. Fourth cut (H:2): Cost = 2 * 3 (v_pieces) = 6. h_pieces = 3. Total minimum cost = 5 + 4 + 9 + 6 = 24.

Common mistakes candidates make

• Not sorting the costs: Trying to cut in order or using a simple average, which leads to a much higher cost.
• Incorrect multiplier: Using the wrong piece counter (e.g., multiplying horizontal costs by horizontal pieces).
• Overcomplicating with DP: While DP works for small constraints, the Greedy approach is $O(M \log M + N \log N)$ and much more efficient.

Interview preparation tip

Remember the rule of thumb for this Sorting interview pattern: "Deal with the most expensive items first." By making the most expensive cuts when there are few pieces, you prevent that high cost from being multiplied across many pieces later.