To solve this problem, we need to compute the combined projection area of cubes on three orthogonal planes: xy, yz, and zx. This requires iterating over the grid to sum the projections for each axis, ensuring an efficient solution with O(N^2) time complexity.
Problem Statement
You are given a grid representing a 3D shape where each value in the grid corresponds to the height of a tower of cubes. The grid is of size n x n, and each value grid[i][j] indicates the height of a tower at position (i, j).
Your goal is to compute the projection area of the shape when viewed from three orthogonal planes: the xy-plane, the yz-plane, and the zx-plane. The total projection area is the sum of the areas projected onto these planes.
Examples
Example 1
Input: grid = [[1,2],[3,4]]
Output: 17
Here are the three projections ("shadows") of the shape made with each axis-aligned plane.
Example 2
Input: grid = [[2]]
Output: 5
Example details omitted.
Example 3
Input: grid = [[1,0],[0,2]]
Output: 8
Example details omitted.
Constraints
- n == grid.length == grid[i].length
- 1 <= n <= 50
- 0 <= grid[i][j] <= 50
Solution Approach
Sum Projections on the xy-Plane
To compute the projection onto the xy-plane, simply sum the number of cubes visible from each cell of the grid. This is done by adding the maximum height in each column.
Sum Projections on the yz-Plane
For the yz-plane, we need to consider the tallest tower in each row. The projection for this plane is the sum of the maximum heights from each row.
Sum Projections on the zx-Plane
The projection onto the zx-plane is calculated similarly to the yz-plane but column-wise. For each column, the maximum height value is added to the total projection area.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(N^2) |
| Space | O(1) |
The time complexity of this solution is O(N^2) because we need to iterate over the entire grid to compute the projections on each plane. The space complexity is O(1) as the solution only uses a constant amount of extra space for intermediate calculations.
What Interviewers Usually Probe
- Candidate can efficiently compute projections in O(N^2) time.
- Candidate uses appropriate methods to calculate max values in rows and columns.
- Candidate avoids unnecessary space usage, demonstrating space optimization.
Common Pitfalls or Variants
Common pitfalls
- Confusing the dimensions of the projections and treating them incorrectly.
- Misunderstanding the projection onto the yz- and zx-planes, treating them the same.
- Failing to handle edge cases, such as grid size 1x1 or the presence of zero heights.
Follow-up variants
- Handle grids of different sizes and edge cases, such as grids with minimum values.
- Extend the problem to higher-dimensional projections, considering additional axes.
- Optimize further for larger grids, testing performance for n up to 50.
How GhostInterview Helps
- GhostInterview helps you break down this problem using systematic projection calculation techniques.
- With GhostInterview, you’ll quickly get feedback on your approach for handling each projection area.
- GhostInterview ensures that you correctly optimize the solution to handle large inputs efficiently.
Topic Pages
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- 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 can I calculate the projection area of a 3D shape?
To calculate the projection area, you sum the maximum heights in each row and column for the yz- and zx-planes, and the number of non-zero elements for the xy-plane.
What is the time complexity of the Projection Area of 3D Shapes problem?
The time complexity is O(N^2) because we must iterate through the grid to compute the projections on each plane.
How do I handle zero heights in the grid for the projection?
Zero heights in the grid should be ignored for the xy-plane projection, as they do not contribute to the area. They still affect the yz- and zx-planes by ensuring that the maximum height is correctly calculated.
What is the space complexity of this problem?
The space complexity is O(1), as the solution only requires constant space for intermediate calculations.
What are the variants of the Projection Area of 3D Shapes problem?
Variants include handling larger grids, optimizing performance for grid sizes up to 50, and extending the problem to higher-dimensional projections.
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