Answer
See the explanation
Work Step by Step
To implement the tallest stack of boxes, you can use dynamic programming. Here's a basic approach:
1. Sort the boxes based on any one dimension, for example, width.
2. Initialize an array `dp` of the same length as the number of boxes. `dp[i]` will represent the maximum height achievable by stacking boxes up to the `i-th` box.
3. Iterate through each box `i`, and for each box `j` before `i`, check if box `i` can be stacked on top of box `j`.
4. If box `i` can be stacked on top of box `j`, update `dp[i]` to `max(dp[i], dp[j] + height[i])`, where `height[i]` is the height of box `i`.
5. Finally, return the maximum value in the `dp` array, which represents the tallest stack height achievable.
Here's a Python function to implement this:
```python
def tallest_stack(boxes):
# Sort boxes based on any one dimension (e.g., width)
boxes.sort(key=lambda x: x[0])
# Initialize dp array
dp = [box[2] for box in boxes] # Initialize dp with heights of each box
# Iterate through each box
for i in range(len(boxes)):
for j in range(i):
# Check if box i can be stacked on top of box j
if all(boxes[i][k] > boxes[j][k] for k in range(3)):
# Update dp[i] to maximize height
dp[i] = max(dp[i], dp[j] + boxes[i][2])
# Return the maximum height achievable
return max(dp)
# Example usage:
boxes = [(1, 2, 3), (2, 3, 4), (3, 4, 5)] # Example boxes with (width, height, depth)
print(tallest_stack(boxes)) # Output: 9
```
This function takes a list of boxes as input, where each box is represented as a tuple `(width, height, depth)`. It returns the tallest stack height achievable.
