Separate Squares I

Problem Description

You are given a 2D integer array squares. Each squares[i] = [xi, yi, li] represents the coordinates of the bottom-left point and the side length of a square parallel to the x-axis. Find the minimum y-coordinate value of a horizontal line such that the total area of the squares above the line equals the total area of the squares below the line. Answers within 10^-5 of the actual answer will be accepted. Squares may overlap. Overlapping areas should be counted multiple times.

Key Insights

• The problem requires finding a horizontal line that divides the area of overlapping squares into two equal parts.
• The total area can be calculated by summing the area of each square.
• A binary search approach can be used to efficiently find the y-coordinate that balances the area above and below the line.
• The challenge lies in correctly calculating the area above and below the line given potential overlaps.

Space and Time Complexity

Time Complexity: O(n log h), where n is the number of squares and h is the height range we are searching through. Space Complexity: O(1), as we are using a constant amount of space for variables.

Solution

To solve the problem, we can use a binary search approach to find the minimum y-coordinate that divides the total area of squares into two equal parts. We will:

1. Calculate the total area of all squares.
2. Use binary search to find a y-coordinate where the area above the line equals the area below it.
3. For each midpoint in the binary search, calculate the area of squares above and below the line by iterating through the list of squares and checking their positions relative to the midpoint.

Code Solutions