The goal of this problem is to find a horizontal line that splits the areas of squares into two equal parts. The solution requires applying binary search over the valid answer space, making use of efficient data structures like segment trees for area calculations.
Problem Statement
You are given a list of squares represented by their bottom-left coordinates and side lengths. Each square is aligned parallel to the x-axis. Your task is to find the minimum y-coordinate of a horizontal line that splits the total area covered by the squares into two equal parts: one above the line and the other below.
The total area of the squares above and below the line should be as close as possible. The solution must be precise, with answers within 10^-5 of the correct answer being accepted.
Examples
Example 1
Input: squares = [[0,0,1],[2,2,1]]
Output: 1.00000
Any horizontal line between y = 1 and y = 2 results in an equal split, with 1 square unit above and 1 square unit below. The minimum y-value is 1.
Example 2
Input: squares = [[0,0,2],[1,1,1]]
Output: 1.00000
Since the blue square overlaps with the red square, it will not be counted again. Thus, the line y = 1 splits the squares into two equal parts.
Constraints
- 1 <= squares.length <= 5 * 104
- squares[i] = [xi, yi, li]
- squares[i].length == 3
- 0 <= xi, yi <= 109
- 1 <= li <= 109
- The total area of all the squares will not exceed 1015.
Solution Approach
Binary Search over the Answer Space
Binary search is used over the possible y-coordinate values, narrowing down the range where the area split is balanced. The challenge is to efficiently find the point where the areas above and below are equal.
Line Sweep with Segment Tree
A line sweep combined with a segment tree allows for efficient area calculation as the horizontal line moves. Segment trees help track the areas dynamically as squares are considered for inclusion above or below the line.
Handling Overlaps
Overlapping squares must be carefully handled since their area should only be counted once. This adds complexity when calculating the area above or below the line.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the binary search iterations and the segment tree operations. For each binary search step, we perform a segment tree update/query operation, leading to a complexity of O(log N) per binary search iteration, where N is the number of squares.
What Interviewers Usually Probe
- Can the candidate explain why binary search is effective in this problem?
- Does the candidate know how to efficiently manage area updates when using segment trees?
- Is the candidate aware of the potential pitfalls with overlapping squares?
Common Pitfalls or Variants
Common pitfalls
- Not handling square overlaps correctly, resulting in inaccurate area calculations.
- Incorrect binary search boundaries leading to suboptimal or incorrect solutions.
- Failure to efficiently use the segment tree for area updates during the line sweep process.
Follow-up variants
- What if the number of squares increases significantly? Can the solution scale to handle larger inputs?
- What happens if we modify the problem to split the area into a ratio other than 50/50?
- How would the solution change if we were to account for square rotation or other transformations?
How GhostInterview Helps
- GhostInterview guides you through a structured approach to solve this problem by focusing on binary search and efficient data structures.
- It helps you understand how to implement a line sweep combined with segment trees for dynamic area calculations.
- The platform provides hints on optimizing performance and handling overlaps, key to solving the problem.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the core algorithm used in the Separate Squares II problem?
The core algorithm involves binary search over the valid answer space, combined with a line sweep and segment tree for efficient area calculations.
How does binary search help in solving Separate Squares II?
Binary search helps by narrowing down the y-coordinate value where the areas above and below the line are equal. It reduces the search space effectively.
Why is a segment tree used in this problem?
A segment tree is used to efficiently calculate and update the area covered by squares as the line sweeps across the y-axis.
What should be the accuracy of the solution in Separate Squares II?
The solution must be accurate to within 10^-5 of the correct y-coordinate for the area split.
How can overlapping squares affect the solution in Separate Squares II?
Overlapping squares must be handled carefully to ensure their area is not double-counted, which can affect the accuracy of the area split.
Need direct help with Separate Squares II instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Separate Squares II from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Determine the number of fruit types that remain unplaced after all allocations in the "Fruits Into Baskets II" problem.
Open problem page#3479 Fruits Into Baskets IIIFruits Into Baskets III requires placing fruits into baskets efficiently using binary search over the answer space for correct allocation.
Open problem page#3501 Maximize Active Section with Trade IIMaximize the number of active sections in a binary string with at most one trade.
Open problem page