To solve the Minimum Area Rectangle problem, scan through the points and utilize a hash table for efficient lookups. For each pair of points, check if their other diagonal corners exist in the set of points, forming a rectangle. Return the smallest possible area. If no rectangle can be formed, return 0.
Problem Statement
You are given an array of points on a 2D plane, where each point is represented as a pair of integers. You need to find the minimum area of a rectangle formed by these points with sides parallel to the X and Y axes. If no such rectangle can be formed, return 0.
For example, given the points [[1,1],[1,3],[3,1],[3,3],[2,2]], a rectangle can be formed between the points (1,1), (1,3), (3,1), and (3,3). The minimum area of the rectangle is 4. Your solution should consider efficient strategies for finding the rectangle with minimal area.
Examples
Example 1
Input: points = [[1,1],[1,3],[3,1],[3,3],[2,2]]
Output: 4
Example details omitted.
Example 2
Input: points = [[1,1],[1,3],[3,1],[3,3],[4,1],[4,3]]
Output: 2
Example details omitted.
Constraints
- 1 <= points.length <= 500
- points[i].length == 2
- 0 <= xi, yi <= 4 * 104
- All the given points are unique.
Solution Approach
Array Scanning and Hash Lookup
First, scan through all point pairs, checking if two points can form a diagonal of a rectangle. Use a hash table to store all points for fast lookups. If both diagonal points are found, calculate the area of the rectangle.
Efficient Rectangle Area Calculation
For each valid rectangle, calculate the area using the formula (width * height) and track the minimum area. Update the result when a smaller area is found.
Avoiding Duplicate Rectangles
Since the points are unique, you don't need to worry about duplicates. However, make sure to consider only the pairs that form valid rectangles, avoiding any unnecessary checks for invalid configurations.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the approach, but an optimized solution with hash lookups and array scanning can bring it down to O(n^2), where n is the number of points. Space complexity is O(n) due to the hash table used for fast lookups.
What Interviewers Usually Probe
- Assessing the candidate's understanding of hashing and geometric properties of the points.
- Check if the candidate efficiently handles the edge case of no valid rectangle being possible.
- Determine if the candidate can optimize time complexity and avoid unnecessary computations.
Common Pitfalls or Variants
Common pitfalls
- Failing to correctly check for diagonal corners forming a valid rectangle.
- Overcomplicating the solution by not using hash lookups to speed up the search.
- Not considering the case where no rectangle can be formed, which should return 0.
Follow-up variants
- Finding the largest area rectangle instead of the smallest.
- Handling different sets of points with varying coordinates for efficiency.
- Implementing the algorithm with a different data structure (e.g., sorted arrays instead of hash tables).
How GhostInterview Helps
- GhostInterview assists by breaking down the problem into an array scanning and hash lookup approach, guiding the candidate to an optimal solution.
- The solver suggests handling edge cases, like when no rectangle exists, helping candidates address all potential scenarios.
- GhostInterview's example walkthrough ensures the candidate practices efficiently using hash tables for point lookups, optimizing both time and space.
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 minimum area rectangle problem in LeetCode?
The Minimum Area Rectangle problem asks you to find the smallest rectangle formed by given points on a 2D plane, with sides parallel to the axes.
How does array scanning help in the Minimum Area Rectangle problem?
Array scanning helps by iterating over pairs of points to identify possible diagonal corners of a rectangle, optimizing the search process.
What is the role of hash lookup in the Minimum Area Rectangle problem?
Hash lookup speeds up the process of checking if the other diagonal corners of a rectangle exist, reducing the time complexity of the solution.
How do you avoid unnecessary computations in this problem?
By using a hash table to check for valid rectangle corners, you can avoid repeatedly scanning all points, improving efficiency.
What is the time complexity of the Minimum Area Rectangle problem?
The optimized time complexity is O(n^2), where n is the number of points, due to the need to check pairs of points for potential diagonals.
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