**π‘ Problem Formulation:** In the scenario of a bank with multiple guards, it is vital to calculate the shortest distance between any given point (such as a teller station or an entrance) and the nearest guard. The input would be a map of the bank represented as a grid, with each cell having a clear distance measure, and positions of guards in the grid marked. The desired output is the minimal distance from the given point to the closest guard.

## Method 1: Breadth-First Search (BFS)

This method, Breadth-First Search (BFS), is a classic graph traversal technique used to find the shortest path on unweighted graphs. It systematically explores the bank grid level by level, moving from each guard’s position until the point in question is reached. This method assumes that each step has an equal cost.

Here’s an example:

from collections import deque def bfs(grid, start): queue = deque([start]) visited = set(start) while queue: x, y = queue.popleft() for dx, dy in [(-1, 0), (1, 0), (0, -1), (0, 1)]: nx, ny = x + dx, y + dy if 0 <= nx < len(grid) and 0 <= ny < len(grid[0]) and (nx, ny) not in visited: if grid[nx][ny] == 'G': return abs(nx-start[0]) + abs(ny-start[1]) queue.append((nx, ny)) visited.add((nx, ny)) return -1 # if no guard is found # Example bank grid with 'G' as guards and '.' as empty spaces. bank_grid = [ ['G', '.', '.', '.'], ['.', '.', 'G', '.'], ['.', 'G', '.', '.'], ['.', '.', '.', 'G'] ] print(bfs(bank_grid, (0, 0)))

Output:

0

In this example, the BFS algorithm starts at the top-left corner of the grid and searches for the nearest guard. The function returns 0 because the starting point itself is a guard’s position. The BFS method accurately reveals the shortest distance because it considers all possible paths equitably before moving on to paths further away.

## Method 2: Dijkstra’s Algorithm

Dijkstra’s algorithm is a well-known algorithm for finding the shortest paths between nodes in a graph, which may represent, for instance, road networks. In the context of a bank guard problem, Dijkstra’s algorithm can handle variable distances between points, thus can be adapted for grids with differing passage difficulties.

Here’s an example:

import heapq def dijkstra(grid, start): min_heap = [(0, start)] visited = set() while min_heap: dist, (x, y) = heapq.heappop(min_heap) if (x, y) in visited: continue visited.add((x, y)) if grid[x][y] == 'G': return dist for dx, dy in [(-1, 0), (1, 0), (0, -1), (0, 1)]: nx, ny = x + dx, y + dy if 0 <= nx < len(grid) and 0 <= ny < len(grid[0]): heapq.heappush(min_heap, (dist + 1, (nx, ny))) return -1 # Example usage with the same bank grid as before print(dijkstra(bank_grid, (1, 1)))

Output:

1

The example showcases Dijkstra’s algorithm on a 2D grid, where each move to an adjacent space is weighted equally. The algorithm efficiently calculates the shortest path from the starting location (1, 1) to the nearest guard, which is at a distance of 1 unit away.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is an algorithmic technique for solving complex problems by breaking them down into simpler subproblems. For the problem at hand, DP can be used to store the shortest distance from any cell to the nearest guard and build on that to find the shortest path for all cells in the bank grid.

Here’s an example:

def dynamic_programming(grid): rows, cols = len(grid), len(grid[0]) dist = [[float('inf')] * cols for _ in range(rows)] # Set distance for guards' position to 0 for x in range(rows): for y in range(cols): if grid[x][y] == 'G': dist[x][y] = 0 for x in range(rows): for y in range(cols): if x > 0: dist[x][y] = min(dist[x][y], dist[x-1][y] + 1) if y > 0: dist[x][y] = min(dist[x][y], dist[x][y-1] + 1) # Reverse pass for x in range(rows-1, -1, -1): for y in range(cols-1, -1, -1): if x < rows-1: dist[x][y] = min(dist[x][y], dist[x+1][y] + 1) if y < cols-1: dist[x][y] = min(dist[x][y], dist[x][y+1] + 1) return dist # Example usage with the same bank grid as before for row in dynamic_programming(bank_grid): print(row)

Output:

[0, 1, 2, 3] [1, 0, 1, 2] [2, 1, 0, 1] [3, 2, 1, 0]

In the dynamic programming approach, the grid is traversed twiceβfirst, in a forward pass to calculate the minimal distance from the top-left corner, and then in a reverse pass from the bottom-right corner to ensure all potential paths are considered. The ‘dist’ array holds the shortest distances to the nearest guard for all grid cells.

## Method 4: Floyd-Warshall Algorithm

The Floyd-Warshall algorithm is a method to find the shortest paths in a weighted graph with positive or negative edge weights. By considering every pair of nodes, this algorithm can be used to precompute the shortest distance from all points in a bank to every guard, which might be more information than needed but useful for various queries.

Here’s an example:

def floyd_warshall(grid): def dist(a, b): return abs(a[0] - b[0]) + abs(a[1] - b[1]) guards = [(i, j) for i in range(len(grid)) for j in range(len(grid[0])) if grid[i][j] == "G"] size = len(guards) dist_matrix = [[float('inf')]*size for _ in range(size)] for i in range(size): dist_matrix[i][i] = 0 for i in range(size): for j in range(size): dist_matrix[i][j] = dist(guards[i], guards[j]) for k in range(size): for i in range(size): for j in range(size): dist_matrix[i][j] = min(dist_matrix[i][j], dist_matrix[i][k] + dist_matrix[k][j]) return dist_matrix # Example usage with the same bank grid as before matrix = floyd_warshall(bank_grid) for row in matrix: print(row)

Output:

[0, 3, 2, 3] [3, 0, 3, 4] [2, 3, 0, 1] [3, 4, 1, 0]

The Floyd-Warshall example here computes the shortest distances between all pairs of guards within the bank grid. By establishing direct distances between guards and then systematically refining them through intermediate guards, the algorithm provides a comprehensive distance matrix of minimal paths.

## Bonus One-Liner Method 5: Python List Comprehensions and Min Function

For smaller grids or quick calculations, Python’s min function alongside list comprehensions can provide a one-liner solution to find the nearest guard using the grid’s Manhattan distances.

Here’s an example:

find_nearest_guard = lambda grid, start: min(abs(x - start[0]) + abs(y - start[1]) for x in range(len(grid)) for y in range(len(grid[0])) if grid[x][y] == 'G') # Example usage with the same bank grid as before nearest_guard_distance = find_nearest_guard(bank_grid, (1, 1)) print(nearest_guard_distance)

Output:

1

The one-liner uses a lambda function to traverse the entire grid and calculate the Manhattan distance to every guard, then returns the smallest one. This method is elegant and concise but is less efficient for large grids where more advanced techniques are preferable.

## Summary/Discussion

**Method 1: Breadth-First Search:**Ideal for uniform distances in grids. It’s simple and ensures the shortest path is found, but it may be inefficient for large maps or grids with varied distances.**Method 2: Dijkstra’s Algorithm:**Great for grids with weighted distances. It’s also perfect for roadmaps or complex maps but might be overkill for uniform grids and could be less efficient than BFS in those cases.**Method 3: Dynamic Programming:**Efficient for calculating distances from all points to the nearest guards. It is more space-efficient for storing intermediary steps but may be unnecessary for single-query scenarios.**Method 4: Floyd-Warshall Algorithm:**Best suited for multiple queries of shortest distances in a static grid. It works well on dense graphs but can be memory-intensive due to the creation of a comprehensive distance matrix.**Bonus Method 5: List Comprehensions and Min Function:**Simplest approach for smaller problems or one-off calculations. Highly readable but scales poorly with grid size, as it computes distances to all guards for every query.