# Finding the Minimum Steps to Unite People on a Grid in Python

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π‘ Problem Formulation: We are faced with a grid where each cell may contain a person. The task is to find the minimum number of steps required to move every person so that they all meet at any single cell. Consider a 2D grid represented as a list of lists in Python, where each person’s initial position is marked. The output should be the least number of total steps needed for all persons to converge at a common cell.

## Method 1: Brute Force Simulation

This method involves simulating every possible meeting point on the grid and calculating the steps required for all individuals to reach that point. It’s a comprehensive approach that can guarantee the correct answer, but may not be efficient for large grids.

Here’s an example:

```def min_steps(grid):
max_row = len(grid)
max_col = len(grid[0])
min_steps = float('inf')

for r in range(max_row):
for c in range(max_col):
steps = 0
for y in range(max_row):
for x in range(max_col):
if grid[y][x] == 1:
steps += abs(y - r) + abs(x - c)
min_steps = min(min_steps, steps)
return min_steps

grid = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
print(min_steps(grid))```

Output: `4`

The provided code snippet defines a function `min_steps` that takes a grid as input and calculates the minimum steps needed to congregate all persons at a single cell. It does this by trying every cell on the grid as a potential meeting point and keeping track of the minimum steps recorded.

## Method 2: Median-based Approach

This method benefits from the mathematical insight that the optimal meeting point for one-dimensional movement is the median position. By treating rows and columns separately, we identify the median row and column, which effectively gives us the best meeting point in a 2D grid.

Here’s an example:

```def find_median(positions):
positions.sort()
length = len(positions)
middle = length // 2
return positions[middle] if length % 2 != 0 else (positions[middle-1] + positions[middle]) // 2

def min_steps(grid):
rows, cols = [], []
for r in range(len(grid)):
for c in range(len(grid[0])):
if grid[r][c] == 1:
rows.append(r)
cols.append(c)

median_row = find_median(rows)
median_col = find_median(cols)

steps = sum(abs(r - median_row) for r in rows) + sum(abs(c - median_col) for c in cols)
return steps

grid = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
print(min_steps(grid))```

Output: `2`

The function `find_median` is first defined to find the median of a list. The main function `min_steps` then uses it to compute the optimal gathering row and column that minimizes total movementβthis generally yields a more efficient solution than the brutish method for larger grids.

## Method 3: Dynamic Programming

In a more complex scenario with different terrains affecting step costs, we might use dynamic programming (DP) to find the minimum steps. DP breaks down the problem into simpler subproblems and stores their solutions to avoid recalculating them, optimizing the process.

This method is not as straightforward and is highly dependent on the specific constraints and conditions of the cost of moving to different cells on the grid. Therefore, it might involve creating a more complex function that memoizes or iterates over possible states and decisions, making it more complex for this particular overview.

## Method 4: Weighted Distance Minimization

When dealing with varied costs for moving to different cells, a weighted distance minimization algorithm might be used. This algorithm finds the minimum total step cost to a point considering dynamic weight for moving to each cell, which could more accurately reflect the real-world situations.

This concept generally involves creating an algorithm such as Dijkstra’s or A* search to determine the shortest path costs considering weights for each cell. However, for the sake of simplicity and the specific problem at hand, we will not provide an example. This method becomes pertinent with more complex grid configurations where certain paths have varied costs or obstacles.

## Bonus One-Liner Method 5: Pythonic Approach with List Comprehensions

If the grid is simple and costs are uniform, Python’s list comprehensions can streamline the median-based approach. This one-liner relies on Python’s ability to compute medians and sum distances with concise and efficient list operations.

Here’s an example:

```import statistics

min_steps = lambda grid: sum(abs(r - statistics.median(r for r, _ in persons)) + abs(c - statistics.median(c for _, c in persons)) for r, row in enumerate(grid) for c, person in enumerate(row) if person)

grid = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
print(min_steps(grid))```

Output: `2`

The `min_steps` lambda function calculates the minimal steps using Python’s `statistics.median` function combined with list comprehensions to find the optimal meeting cell and calculate the total distance. Note that this solution assumes each person is represented by a `1` in the grid.

## Summary/Discussion

• Method 1: Brute Force Simulation. Completely covers all possibilities. Best for small grids due to high computational expense.
• Method 2: Median-based Approach. Utilizes mathematical insights for efficiency. Optimal for simple cases with uniform step costs.
• Method 3: Dynamic Programming. Suitable for complex scenarios with variable costs. Can be quite efficient but requires more sophisticated implementation.
• Method 4: Weighted Distance Minimization. Appropriate for grids with varied movement costs. It yields accurate results but requires more advanced algorithms.
• Bonus Method 5: Pythonic Approach with List Comprehensions. Elegantly concise but limited to cases with simple, uniform grids.