#1465

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts

Maximize the area of a piece of cake after specified horizontal and vertical cuts using greedy algorithms and sorting.

Problem Statement

You are given a rectangular cake with dimensions h x w. Two arrays, horizontalCuts and verticalCuts, represent the positions at which the cake is cut horizontally and vertically. Your task is to compute the maximum area of any piece of cake after performing all cuts at the specified positions in these arrays.

The area of the largest piece of cake must be returned modulo 10^9 + 7, as the value can be quite large. You must efficiently determine the maximum piece area by evaluating the differences between adjacent cuts and computing the product of the maximum horizontal and vertical cut differences.

Examples

Example 1

Input: h = 5, w = 4, horizontalCuts = [1,2,4], verticalCuts = [1,3]

Output: 4

The figure above represents the given rectangular cake. Red lines are the horizontal and vertical cuts. After you cut the cake, the green piece of cake has the maximum area.

Example 2

Input: h = 5, w = 4, horizontalCuts = [3,1], verticalCuts = [1]

Output: 6

The figure above represents the given rectangular cake. Red lines are the horizontal and vertical cuts. After you cut the cake, the green and yellow pieces of cake have the maximum area.

Example 3

Input: h = 5, w = 4, horizontalCuts = [3], verticalCuts = [3]

Output: 9

Example details omitted.

Constraints

2 <= h, w <= 109
1 <= horizontalCuts.length <= min(h - 1, 105)
1 <= verticalCuts.length <= min(w - 1, 105)
1 <= horizontalCuts[i] < h
1 <= verticalCuts[i] < w
All the elements in horizontalCuts are distinct.
All the elements in verticalCuts are distinct.

Solution Approach

Sorting the Cuts

First, sort the horizontalCuts and verticalCuts arrays. Sorting helps identify the largest gaps between cuts, which will lead to the largest piece area. Sorting is crucial for simplifying the problem of finding the largest piece after all cuts.

Calculating Maximum Gaps

After sorting, calculate the largest differences between consecutive elements in the horizontalCuts and verticalCuts arrays. This step helps find the maximum gap between cuts, which directly translates to the largest possible dimensions of the cake pieces.

Multiplying Maximum Gaps

Finally, multiply the maximum horizontal gap by the maximum vertical gap to compute the largest piece area. This result is then taken modulo 10^9 + 7 to handle large numbers.

Complexity Analysis

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

The time complexity is dominated by the sorting step, which takes O(n log n) for horizontal and vertical cuts. The space complexity is O(n) due to the storage of sorted arrays, where n is the maximum length of the cuts arrays.

What Interviewers Usually Probe

Does the candidate recognize that sorting is a key step?
Can the candidate identify the greedy choice behind the problem?
Does the candidate handle the modulo operation correctly when computing the area?

Common Pitfalls or Variants

Common pitfalls

Forgetting to sort the cuts, which leads to incorrect gap calculations.
Overcomplicating the solution by trying to evaluate all possible sub-areas, instead of focusing on maximum gap differences.
Neglecting the modulo operation, causing overflow issues when the area is large.

Follow-up variants

What happens if the cuts are in descending order? The sorting step will still handle it correctly.
How does this solution scale when the number of cuts is very large? The time complexity remains efficient due to sorting.
What if there are no vertical or horizontal cuts? The maximum area would be the full area of the cake.

FAQ

How do I approach the 'Maximum Area of a Piece of Cake' problem?

Sort the horizontal and vertical cuts, then compute the largest differences between adjacent cuts. Multiply these maximum differences to get the maximum area of a piece of cake.

What is the pattern for solving 'Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts'?

This problem uses the greedy choice pattern, where we optimize by selecting the largest gaps between cuts, ensuring maximum area.

Why is sorting the cuts important in this problem?

Sorting helps identify the largest gaps between cuts, which are crucial for maximizing the area of the resulting pieces.

How do I ensure the solution handles large numbers in the 'Maximum Area of a Piece of Cake' problem?

By returning the result modulo 10^9 + 7, you ensure the area remains within a manageable range, even for large inputs.

What happens if there are no vertical or horizontal cuts?

If no cuts are made, the entire cake remains intact, and the area of the cake is simply the product of its dimensions.

terminal

Solution

Solution 1: Sorting

We first sort `horizontalCuts` and `verticalCuts` separately, and then traverse both arrays to calculate the maximum difference between adjacent elements. We denote these maximum differences as $x$ and $y$, respectively. Finally, we return $x \times y$.

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class Solution:
    def maxArea(
        self, h: int, w: int, horizontalCuts: List[int], verticalCuts: List[int]
    ) -> int:
        horizontalCuts.extend([0, h])
        verticalCuts.extend([0, w])
        horizontalCuts.sort()
        verticalCuts.sort()
        x = max(b - a for a, b in pairwise(horizontalCuts))
        y = max(b - a for a, b in pairwise(verticalCuts))
        return (x * y) % (10**9 + 7)

Continue Practicing

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