**π‘ Problem Formulation:** Imagine you have a grid of buildings viewed from the top, with each building having its own height. The goal is to increase the heights of the buildings to maximize the new skyline’s increase, without altering the shape of the viewed skyline from any direction. If the input is a 2D list representing a city grid (e.g., `[[3, 0, 8, 4],[2, 4, 5, 7],[9, 2, 6, 3],[0, 3, 1, 0]]`

), the desired output is an integer representing the maximum total sum that the building heights can be increased (e.g., `35`

).

## Method 1: Brute Force Iteration

This method goes through each building in the grid, determining the highest buildings in its row and column, and increases the height to the minimum of these two values. The function specification would include a function that accepts a 2D list and returns an integer representing the total increase.

Here’s an example:

def max_increase_keeping_skyline(grid): total = 0 for i in range(len(grid)): for j in range(len(grid[0])): max_row = max(grid[i]) max_col = max(grid[k][j] for k in range(len(grid))) total += min(max_row, max_col) - grid[i][j] return total print(max_increase_keeping_skyline([[3, 0, 8, 4], [2, 4, 5, 7], [9, 2, 6, 3], [0, 3, 1, 0]]))

Output: `35`

This loop iterates over each building, determines the tallest building in the building’s row and column, then calculates the difference between the minimum of these two values and the original height. The sum of these differences across all buildings gives the maximum increase.

## Method 2: Precomputed Row and Column Maxima

This method first precomputes the maximum height of each row and column. Then, it iterates over the buildings in the grid, computing the minimum between the row’s and column’s maximum heights and increments the total increase. It is more efficient as it does not recalculate maxima for each building.

Here’s an example:

def max_increase_keeping_skyline(grid): row_maxes = [max(row) for row in grid] col_maxes = [max(col) for col in zip(*grid)] return sum(min(row_maxes[i], col_maxes[j]) - val for i, row in enumerate(grid) for j, val in enumerate(row)) print(max_increase_keeping_skyline([[3, 0, 8, 4], [2, 4, 5, 7], [9, 2, 6, 3], [0, 3, 1, 0]]))

Output: `35`

The code starts by creating lists of maximum heights for rows and columns. It then uses list comprehension to calculate the sum of increases across the city grid, ensuring the skyline’s shape remains the same.

## Method 3: Using NumPy to Optimize Array Operations

In this method, Python’s NumPy library is leveraged for its efficient array operations. The maximum height for each row and column is calculated using NumPy’s `max`

function, and NumPy’s broadcasting feature is used to compute the total increase directly from these maxima.

Here’s an example:

import numpy as np def max_increase_keeping_skyline(grid): grid_np = np.array(grid) row_maxes = np.max(grid_np, axis=1) col_maxes = np.max(grid_np, axis=0) total_increase = np.sum(np.minimum(row_maxes[:, np.newaxis], col_maxes) - grid_np) return total_increase print(max_increase_keeping_skyline([[3, 0, 8, 4], [2, 4, 5, 7], [9, 2, 6, 3], [0, 3, 1, 0]]))

Output: `35`

NumPy’s efficient array operations make this code more concise and likely faster for larger grids. This is because NumPy uses optimized C code under the hood, reducing the computation time significantly.

## Method 4: Dynamic Programming

Dynamic programming is used to solve this problem by storing the maximum heights of rows and columns as the program iterates over the grid. This method also allows for future extensions, such as modifying the skyline or adding new buildings.

Here’s an example:

def max_increase_keeping_skyline(grid): n = len(grid) row_maxes = [0] * n col_maxes = [0] * n for i in range(n): for j in range(n): row_maxes[i] = max(row_maxes[i], grid[i][j]) col_maxes[j] = max(col_maxes[j], grid[i][j]) total = sum(min(row_maxes[i], col_maxes[j]) - grid[i][j] for i in range(n) for j in range(n)) return total print(max_increase_keeping_skyline([[3, 0, 8, 4], [2, 4, 5, 7], [9, 2, 6, 3], [0, 3, 1, 0]]))

Output: `35`

This method stores the maxima in separate lists and avoids unnecessary recalculations, making it efficient and scalable for larger datasets while adhering to dynamic programming principles.

## Bonus One-Liner Method 5: Pythonic List Comprehension with Map and Zip

By using Python’s list comprehension alongside `map`

and `zip`

, this method offers a one-liner solution that is both elegant and Pythonic. It is best for those who prefer succinct code and are comfortable with functional programming paradigms.

Here’s an example:

def max_increase_keeping_skyline(grid): return sum(min(max(row), max(col)) - val for row, col, val in zip(grid, zip(*grid), sum(grid, []))) print(max_increase_keeping_skyline([[3, 0, 8, 4], [2, 4, 5, 7], [9, 2, 6, 3], [0, 3, 1, 0]]))

Output: `35`

This compact Python one-liner uses `zip`

to pair each element of the grid with its corresponding maxima in the row and column dimensions. The total is computed using a sum of the differences, similar to the earlier methods, but condensed into a single, elegant expression.

## Summary/Discussion

**Method 1: Brute Force Iteration.**Simple to understand and implement. Can be slow for large grids due to repeated maxima calculations.**Method 2: Precomputed Row and Column Maxima.**More efficient than brute force. Precomputing maxima reduces redundant calculations. Still not as fast as utilizing specialized libraries.**Method 3: Using NumPy to Optimize Array Operations.**Utilizes optimized NumPy operations. Offers significant speed improvements for large datasets. Requires the additional installation of NumPy.**Method 4: Dynamic Programming.**Scales well with grid size. Optimized through storing maxima. Allows for expandability and adaptability to changes in the problem specification.**Bonus Method 5: Pythonic List Comprehension with Map and Zip.**Elegant and concise. Best for Python enthusiasts and concise code lovers. However, it may not be as readable for those unfamiliar with Python’s functional aspects.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.