To solve this problem, efficiently scan the list of points and use a hash table to count points sharing the same y-coordinate. Check all possible combinations of points forming horizontal trapezoids by comparing the y-coordinates of selected points. Analyzing the problem in terms of geometric properties, such as parallel lines, simplifies the approach significantly.
Problem Statement
You are given a 2D integer array points, where each point is represented as [xi, yi] on the Cartesian plane. A horizontal trapezoid is a convex quadrilateral that has at least one pair of sides parallel to the x-axis. This means that the selected points must have the same y-coordinate for the parallel sides.
Your task is to return the number of distinct horizontal trapezoids that can be formed by selecting four points from the array, with the constraint that all points are pairwise distinct.
Examples
Example 1
Input: points = [[1,0],[2,0],[3,0],[2,2],[3,2]]
Output: 3
There are three distinct ways to pick four points that form a horizontal trapezoid:
Example 2
Input: points = [[0,0],[1,0],[0,1],[2,1]]
Output: 1
There is only one horizontal trapezoid that can be formed.
Constraints
- 4 <= points.length <= 105
- –108 <= xi, yi <= 108
- All points are pairwise distinct.
Solution Approach
Scan the points and categorize by y-coordinate
To efficiently count horizontal trapezoids, scan through the array of points, grouping the points by their y-coordinate. This allows easy identification of potential pairs of parallel lines.
Use hash table to track frequency
Implement a hash table to store the frequency of each y-coordinate. By doing so, it becomes easier to calculate how many points lie on a line parallel to the x-axis, which helps to form the parallel sides of the trapezoid.
Count valid trapezoids based on conditions
Once the points are categorized and stored, count the number of ways to select four points with the same y-coordinate to form the two horizontal sides of a trapezoid. Carefully check the geometric conditions for a valid trapezoid formation.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the hash table operations, which in the worst case are O(n) for scanning the points and O(1) for each insertion or lookup. The space complexity is O(n) due to the need to store the points and hash table entries.
What Interviewers Usually Probe
- Candidate demonstrates understanding of geometric properties like parallel lines.
- Candidate efficiently uses hash tables to track and group points.
- Candidate is able to optimize their approach to handle the input size constraint.
Common Pitfalls or Variants
Common pitfalls
- Misunderstanding the concept of parallel lines, leading to incorrect trapezoid formation.
- Failure to account for all distinct point combinations.
- Not optimizing the approach to handle larger inputs within time limits.
Follow-up variants
- Implementing the solution in a higher-dimensional space.
- Extending the solution to count trapezoids with vertical sides as well.
- Exploring alternative approaches like sorting or dynamic programming.
How GhostInterview Helps
- GhostInterview's problem-specific solver assists in identifying efficient approaches to handle large input sizes.
- Helps in leveraging geometric properties, focusing on line parallelism and y-coordinate grouping.
- The tool guides users to avoid common pitfalls and suggests optimal use of hash tables for fast lookups.
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
How do I count the number of horizontal trapezoids efficiently?
Efficiently count trapezoids by scanning through the points and grouping them by their y-coordinate using a hash table. Then, calculate the possible trapezoids from those groups.
What is the primary pattern in this problem?
The primary pattern is array scanning plus hash lookup, which helps categorize points by their y-coordinate to identify potential trapezoids.
Can the approach handle large input sizes?
Yes, the approach is designed to handle inputs up to 10^5 points efficiently by utilizing hash tables for fast lookups.
What role does the y-coordinate play in forming a trapezoid?
The y-coordinate determines which points can form the horizontal sides of the trapezoid, as these sides must be parallel to the x-axis.
How do I optimize the solution for this problem?
Optimize by categorizing points by y-coordinate using a hash table, which enables fast lookups and efficient counting of potential trapezoids.
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Count Number of Trapezoids II requires scanning point pairs and hashing slopes to efficiently find parallel sides in arrays.
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