#3047

Medium

LeetCode Problem Workspace

Find the Largest Area of Square Inside Two Rectangles

Find the largest square area that can fit inside the intersection of two or more rectangles in a 2D plane.

Problem Statement

You are given n rectangles in a 2D plane where each rectangle is defined by its bottom-left and top-right coordinates. You must determine the maximum area of a square that can fit inside the intersection of at least two rectangles. If no square can fit in the intersection, return 0.

The bottom-left and top-right coordinates of each rectangle are provided in two 2D integer arrays, bottomLeft and topRight, respectively. The goal is to find the largest square area that fits in the intersection region of the rectangles, or return 0 if no such square exists.

Examples

Constraints

n == bottomLeft.length == topRight.length
2 <= n <= 103
bottomLeft[i].length == topRight[i].length == 2
1 <= bottomLeft[i][0], bottomLeft[i][1] <= 107
1 <= topRight[i][0], topRight[i][1] <= 107
bottomLeft[i][0] < topRight[i][0]
bottomLeft[i][1] < topRight[i][1]

Solution Approach

Brute Force Approach

Start by checking the intersection of each pair of rectangles. For every possible pair, calculate the overlap and determine the largest square that fits in the intersection. While this method can be slow, it directly calculates the possible intersections.

Optimized Intersection Calculation

Instead of checking every possible pair of rectangles brute-force, look for the common intersection region. Calculate the width and height of the overlap and determine the largest square that fits. This method improves efficiency by narrowing down the search.

Advanced Geometry Methods

Utilize advanced geometry algorithms or computational geometry techniques to quickly find the intersection areas and calculate the largest square that fits. This method is more efficient but requires more complex math and understanding of geometry.

Complexity Analysis

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

The time and space complexity depend on the approach used. The brute force method has a time complexity of O(n^2) for checking each rectangle pair, while optimized methods can improve this. Space complexity depends on the data storage for each rectangle and its intersection, typically O(n).

What Interviewers Usually Probe

Candidate understands how to calculate the intersection of rectangles and how to approach the problem efficiently.
Look for knowledge of computational geometry techniques and optimization strategies for handling large datasets.
The candidate should identify potential trade-offs between brute force and more advanced solutions based on time and space constraints.

Common Pitfalls or Variants

Common pitfalls

Not accounting for the case where no intersection exists, which would return a result of 0.
Ignoring the rectangle boundaries when calculating the intersection, which may result in incorrect overlap calculations.
Using brute force without considering optimization, which can lead to performance issues when handling large inputs.

Follow-up variants

Instead of calculating the largest square, the task could ask for the largest rectangle within the intersection.
The problem could be extended to handle non-rectangular shapes in the intersection, complicating the geometric calculations.
The problem could involve calculating the largest square inside a single rectangle, with no intersection required.

FAQ

What is the main idea of the 'Find the Largest Area of Square Inside Two Rectangles' problem?

The problem asks you to find the largest square area that can fit inside the intersection of at least two rectangles on a 2D plane.

How can brute force be used to solve this problem?

Brute force can be used by checking every pair of rectangles, calculating their intersection, and finding the largest square that fits in that region.

What are the potential performance issues when solving this problem?

The main performance issue arises from brute force methods that check all possible rectangle pairs, leading to a time complexity of O(n^2), which is inefficient for large inputs.

Are there optimization strategies for this problem?

Yes, you can optimize by calculating the intersection region for each pair of rectangles and only focusing on the largest square that fits inside that region.

What happens if no intersection exists between rectangles?

If no intersection exists, the result should be 0, as no square can fit inside the intersecting region.

terminal

Solution

Solution 1: Enumeration

We can enumerate two rectangles, where the coordinates of the bottom left and top right corners of rectangle 1 are $(x_1, y_1)$ and $(x_2, y_2)$ respectively, and the coordinates of the bottom left and top right corners of rectangle 2 are $(x_3, y_3)$ and $(x_4, y_4)$ respectively.

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class Solution:
    def largestSquareArea(
        self, bottomLeft: List[List[int]], topRight: List[List[int]]
    ) -> int:
        ans = 0
        for ((x1, y1), (x2, y2)), ((x3, y3), (x4, y4)) in combinations(
            zip(bottomLeft, topRight), 2
        ):
            w = min(x2, x4) - max(x1, x3)
            h = min(y2, y4) - max(y1, y3)
            e = min(w, h)
            if e > 0:
                ans = max(ans, e * e)
        return ans

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