#3218

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Cost for Cutting Cake I

In this problem, you need to minimize the cost of cutting a cake into 1x1 pieces using vertical and horizontal cuts.

array dynamic programming greedy sorting

Problem Statement

You are given a cake of size m x n that needs to be cut into 1 x 1 pieces. You are also given two arrays, horizontalCut and verticalCut, which represent the positions at which you can cut the cake horizontally and vertically. You want to minimize the cost of cutting the cake. The cost of each cut is equal to the area of the cake being cut.

In each operation, you can choose any piece of cake that is not yet a 1 x 1 square and perform one of the following cuts: a vertical cut or a horizontal cut. Each cut will split the cake into smaller pieces. The objective is to determine the minimum cost for cutting the entire cake into 1 x 1 pieces.

Examples

Example 1

Input: m = 3, n = 2, horizontalCut = [1,3], verticalCut = [5]

Output: 13

The total cost is 5 + 1 + 1 + 3 + 3 = 13 .

Example 2

Input: m = 2, n = 2, horizontalCut = [7], verticalCut = [4]

Output: 15

The total cost is 7 + 4 + 4 = 15 .

Constraints

1 <= m, n <= 20
horizontalCut.length == m - 1
verticalCut.length == n - 1
1 <= horizontalCut[i], verticalCut[i] <= 103

Solution Approach

Sort the cuts

Start by sorting the horizontalCut and verticalCut arrays to ensure cuts happen in the optimal order. Sorting ensures that you always make the largest possible cuts, reducing the total cost of the operation.

Dynamic Programming for State Transition

Use dynamic programming to calculate the minimum cost. The state transition is based on selecting the best cut at each step, while maintaining a record of the cost accumulated up to that point.

Iterate and Compute Cost

Iterate through the sorted horizontal and vertical cuts, calculating the cost of each cut based on the area of the remaining cake. At each step, the total cost is updated to reflect the current state of the cake.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the approach depends on the sorting of the cuts and the dynamic programming solution. Sorting the cuts takes O(m log m + n log n), where m and n are the lengths of the horizontal and vertical cuts, respectively. The space complexity depends on the data structures used for state transitions.

What Interviewers Usually Probe

Candidate demonstrates understanding of sorting techniques.
Candidate can effectively implement dynamic programming for state transitions.
Candidate identifies and applies greedy principles to minimize cost.

Common Pitfalls or Variants

Common pitfalls

Failing to sort the cuts properly and making cuts in the wrong order.
Not understanding how dynamic programming can minimize the cutting cost over multiple steps.
Overcomplicating the solution by not leveraging the correct greedy algorithm.

Follow-up variants

Use a recursive approach to simulate the process instead of dynamic programming.
Optimize the dynamic programming approach for larger inputs.
Explore alternative greedy approaches for different input sizes.

FAQ

What is the primary approach for solving the Minimum Cost for Cutting Cake I problem?

The problem is best solved using dynamic programming with state transitions, ensuring that the cuts are made in the optimal order to minimize cost.

How do I minimize the cost of cutting the cake in this problem?

By sorting both the horizontal and vertical cuts and applying dynamic programming to track the minimum cost, you can achieve the optimal solution.

Why is sorting important in the Minimum Cost for Cutting Cake I problem?

Sorting ensures that cuts are made in the optimal order, with the largest cuts being made first to minimize the total cost.

What is the time complexity of the solution for this problem?

The time complexity is dominated by the sorting of the cuts, which takes O(m log m + n log n), where m and n are the lengths of the horizontal and vertical cuts.

Can this problem be solved using a greedy algorithm?

Yes, a greedy approach is used by making the largest possible cuts first, but dynamic programming is applied to ensure optimal state transitions for cost minimization.

terminal

Solution

Solution 1: Greedy + Two Pointers

For a given position, the earlier you cut, the fewer cuts are needed, so it is clear that positions with higher costs should be cut earlier.

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class Solution:
    def minimumCost(
        self, m: int, n: int, horizontalCut: List[int], verticalCut: List[int]
    ) -> int:
        horizontalCut.sort(reverse=True)
        verticalCut.sort(reverse=True)
        ans = i = j = 0
        h = v = 1
        while i < m - 1 or j < n - 1:
            if j == n - 1 or (i < m - 1 and horizontalCut[i] > verticalCut[j]):
                ans += horizontalCut[i] * v
                h, i = h + 1, i + 1
            else:
                ans += verticalCut[j] * h
                v, j = v + 1, j + 1
        return ans

Continue Practicing

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