Problem Description
You are given an n x n grid where we place some 1 x 1 x 1 cubes that are axis-aligned with the x, y, and z axes. Each value v = grid[i][j] represents a tower of v cubes placed on top of the cell (i, j). We view the projection of these cubes onto the xy, yz, and zx planes. A projection is like a shadow, that maps our 3-dimensional figure to a 2-dimensional plane. We are viewing the "shadow" when looking at the cubes from the top, the front, and the side. Return the total area of all three projections.
Key Insights
- The projection onto the xy-plane corresponds to counting all non-zero values in the grid.
- The projection onto the yz-plane corresponds to the maximum height in each column of the grid.
- The projection onto the zx-plane corresponds to the maximum height in each row of the grid.
- Combining these three projections gives the total area.
Space and Time Complexity
Time Complexity: O(n^2) - We need to iterate through each element in the n x n grid. Space Complexity: O(1) - We are using a constant amount of extra space.
Solution
To solve the problem, we can follow these steps:
- Initialize three variables to keep track of the projections for the xy-plane, yz-plane, and zx-plane.
- Loop through each cell in the grid:
- For the xy-plane projection, count each non-zero cell.
- For the yz-plane projection, keep track of the maximum height for each column.
- For the zx-plane projection, keep track of the maximum height for each row.
- Sum all the projections to get the total area.
This approach uses basic iteration and keeps track of maximum values using arrays for rows and columns, ensuring we efficiently compute the total projection area.