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You are given an n x n grid represented by a 2D array grid. Each value grid[i][j] represents the height of a 3D shape placed on that i-th row and j-th column. We need to return the projection area of these 3D shapes.

The projection area of a 3D shape on the xy-plane, yz-plane, and zx-plane is described as follows:

  1. xy-plane: The number of cells in the grid that contain a positive integer.
  2. yz-plane: The maximum value in each row.
  3. zx-plane: The maximum value in each column.

The total projection area is the sum of the areas on the xy-plane, yz-plane, and zx-plane.

Clarifying Questions

  1. Is the input always a valid n x n grid?
    • Yes, you can assume the input is always a valid n x n grid.
  2. What range of values can grid[i][j] take?
    • Each value in the grid is a non-negative integer.
  3. How large can n be?
    • Typically, n can be up to 50 in competitive programming scenarios on LeetCode.

Strategy

To solve this problem efficiently, we will iterate through the grid once to compute the areas for the xy-plane, yz-plane, and zx-plane:

  1. xy-plane: Count the number of positive integers in the grid.
  2. yz-plane: For each row, find the maximum value.
  3. zx-plane: For each column, find the maximum value.

Code

def projectionArea(grid):
    n = len(grid)
    xy_area = 0  # Projection on xy-plane
    yz_area = 0  # Projection on yz-plane (rows)
    zx_area = 0  # Projection on zx-plane (columns)
    
    # To compute zx_area, we need to store the max values for each column
    max_in_columns = [0] * n
    
    for i in range(n):
        max_in_row = 0
        for j in range(n):
            height = grid[i][j]
            
            if height > 0:
                xy_area += 1
            
            max_in_row = max(max_in_row, height)
            max_in_columns[j] = max(max_in_columns[j], height)
        
        yz_area += max_in_row
    
    zx_area = sum(max_in_columns)
    
    return xy_area + yz_area + zx_area

Time Complexity

The overall time complexity of this solution is O(n^2) because we are iterating through each cell of the grid once (a total of n^2 cells).

Explanation

  1. xy-plane area: We count each grid cell with a positive integer.
  2. yz-plane area: We find the maximum height in each row and accumulate these values.
  3. zx-plane area: We find the maximum height in each column and sum them up.

By summing these three areas, we get the total projection area of the 3D shapes from all perspectives.

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