Problem Statement
In this scenario, you are provided with a square matrix (grid), where each cell can represent either a thief (denoted as 1) or an empty space (denoted as 0). Starting from the top-left corner of this grid, you can move to any directly adjacent cell. This includes cells that might contain a thief. The main objective is to determine a path that leads to the bottom-right corner of the grid and maximizes the “safeness factor”.
The safeness factor for any path is defined by the smallest Manhattan distance between any cell on that path and any thief on the grid. The Manhattan distance between two points (a, b) and (x, y) is calculated as the absolute difference in their x-coordinates plus the absolute difference in their y-coordinates, represented mathematically as |a - x| + |b - y|.
Your task is to compute the maximum safeness factor among all potential paths from the start to the end of the grid.
Examples
Example 1
Input:
grid = [[1,0,0],[0,0,0],[0,0,1]]
Output:
0
Explanation:
All paths from (0, 0) to (n - 1, n - 1) go through the thieves in cells (0, 0) and (n - 1, n - 1).
Example 2
Input:
grid = [[0,0,1],[0,0,0],[0,0,0]]
Output:
2
Explanation:
The path depicted in the picture above has a safeness factor of 2 since: - The closest cell of the path to the thief at cell (0, 2) is cell (0, 0). The distance between them is | 0 - 0 | + | 0 - 2 | = 2. It can be shown that there are no other paths with a higher safeness factor.
Example 3
Input:
grid = [[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]]
Output:
2
Explanation:
The path depicted in the picture above has a safeness factor of 2 since: - The closest cell of the path to the thief at cell (0, 3) is cell (1, 2). The distance between them is | 0 - 1 | + | 3 - 2 | = 2. - The closest cell of the path to the thief at cell (3, 0) is cell (3, 2). The distance between them is | 3 - 3 | + | 0 - 2 | = 2. It can be shown that there are no other paths with a higher safeness factor.
Constraints
1 <= grid.length == n <= 400grid[i].length == ngrid[i][j]is either0or1.- There is at least one thief in the
grid.
Approach and Intuition
Given the nature of calculating safeness, we can infer the following about optimizing our approach:
Identification of all thieves’ positions:
- By scanning through the grid and recording the positions of all thieves, we can use these positions later to quickly compute distances to any cell on a path.
Dynamic travel through the grid:
- Initiate at cell
(0, 0)and aim to reach(n-1, n-1)considering each cell's adjacency rules.
- Initiate at cell
Calculating Manhattan distances:
- For each cell on a candidate path, calculate the Manhattan distance to each thief and determine the minimum of these distances.
Dynamic Programming (DP) Table setup:
- Use a DP table to store the maximum safeness factor at each cell up to
(n-1, n-1). - Initialize the start point value based on its distance to the nearest thief.
- For other cells, calculate possible safeness factors from each possible prior position (e.g., from left or top) and keep the path which minimizes potential thief proximity while maximizing the path's overall safeness factor.
- Use a DP table to store the maximum safeness factor at each cell up to
Iterative calculation and comparison:
- As you calculate safeness factors through the DP approach, continually compare to maximize the safeness of each cell up to the destination cell
(n-1, n-1).
- As you calculate safeness factors through the DP approach, continually compare to maximize the safeness of each cell up to the destination cell
By following these steps judiciously, the output will provide the maximum possible safeness factor for reaching from the top-left to the bottom-right of the grid, thereby optimally solving the problem within the given constraints.
Solutions
- C++
- Java
- Python
class Solution {
public:
int calculateMaxSafety(vector<vector<int>>& cityMap) {
int n = cityMap.size();
queue<pair<int, int>> bfsQueue;
// Identify thief locations as 0 and mark others as -1 then enqueue thieves
for (int row = 0; row < n; row++) {
for (int column = 0; column < n; column++) {
if (cityMap[row][column] == 1) {
bfsQueue.push({row, column});
cityMap[row][column] = 0;
} else {
cityMap[row][column] = -1;
}
}
}
// Breadth-first search to calculate safety level
while (!bfsQueue.empty()) {
int currentLevelCount = bfsQueue.size();
while (currentLevelCount-- > 0) {
auto current = bfsQueue.front();
bfsQueue.pop();
int currentVal = cityMap[current.first][current.second];
// Process adjacent cells
for (auto& direction : directions) {
int newRow = current.first + direction[0];
int newColumn = current.second + direction[1];
if (cellWithinBounds(cityMap, newRow, newColumn) && cityMap[newRow][newColumn] == -1) {
cityMap[newRow][newColumn] = currentVal + 1;
bfsQueue.push({newRow, newColumn});
}
}
}
}
priority_queue<vector<int>> maxSafetyQueue;
maxSafetyQueue.push({cityMap[0][0], 0, 0});
cityMap[0][0] = -1;
while (!maxSafetyQueue.empty()) {
auto top = maxSafetyQueue.top();
maxSafetyQueue.pop();
if (top[1] == n - 1 && top[2] == n - 1) {
return top[0];
}
for (auto& direction : directions) {
int newRow = direction[0] + top[1];
int newColumn = direction[1] + top[2];
if (cellWithinBounds(cityMap, newRow, newColumn) && cityMap[newRow][newColumn] != -1) {
maxSafetyQueue.push({min(top[0], cityMap[newRow][newColumn]), newRow, newColumn});
cityMap[newRow][newColumn] = -1;
}
}
}
return -1;
}
private:
vector<vector<int>> directions = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
bool cellWithinBounds(vector<vector<int>>& matrix, int row, int col) {
int dim = matrix.size();
return row >= 0 && col >= 0 && row < dim && col < dim;
}
};
The C++ solution provided solves the problem of finding the safest path in a grid. The grid represents a city map where some cells contain threats, and the aim is to calculate the safest path from the start to the destination by avoiding these threats as much as possible.
Here's an approach the code takes:
- Define a class
Solutionwith the methodcalculateMaxSafetywhich accepts acityMaprepresented by a 2D vector. - Use a breadth-first search (BFS) to calculate safety levels from the threats in the grid:
- Initialize a queue to process each cell in a level-wise manner.
- Treat cells with threats (denoted as
1) as starting points and mark them with a safety level of0. - For other cells (no threats initially detected), mark them as
-1. - For each threat cell, explore all adjacent cells and update their safety value based on the proximity to a threat.
- Use a priority queue to determine the safest path from the city map's top-left corner to the bottom-right:
- Prioritize paths based on the maximum safety values of explored cells.
- Traverse the grid based on the calculated safety levels.
- Update the route in the priority queue by comparing the current safety value with the new cell's safety and take the minimum.
- Ensure movement within grid bounds using the helper method
cellWithinBounds, which checks if a given cell coordinate is valid. - Return the safety level of the safest path if such a path exists; otherwise, indicate failure to find a safe route with
-1.
This efficiently implements both safety calculation and pathfinding using BFS and priority queue mechanisms, leveraging effective checks and logical data structure management for optimal problem-solving.
class Solution {
// Movement directions: right, left, down, up
final int[][] directions = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
public int calculateMaxSafety(List<List<Integer>> matrix) {
int size = matrix.size();
int[][] safetyGrid = new int[size][size];
Queue<int[]> processingQueue = new LinkedList<>();
// Convert list to array and initialize queue and grid
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
if (matrix.get(i).get(j) == 1) {
processingQueue.add(new int[] {i, j});
safetyGrid[i][j] = 0;
} else {
safetyGrid[i][j] = -1;
}
}
}
// BFS to calculate safety factor
while (!processingQueue.isEmpty()) {
int count = processingQueue.size();
while (count-- > 0) {
int[] current = processingQueue.poll();
for (int[] direction : directions) {
int newX = current[0] + direction[0];
int newY = current[1] + direction[1];
if (cellIsValid(safetyGrid, newX, newY) && safetyGrid[newX][newY] == -1) {
safetyGrid[newX][newY] = safetyGrid[current[0]][current[1]] + 1;
processingQueue.add(new int[] {newX, newY});
}
}
}
}
// PQ to find max path safety
PriorityQueue<int[]> maxSafetyQueue = new PriorityQueue<>((a, b) -> b[2] - a[2]);
maxSafetyQueue.add(new int[] {0, 0, safetyGrid[0][0]});
safetyGrid[0][0] = -1;
while (!maxSafetyQueue.isEmpty()) {
int[] current = maxSafetyQueue.poll();
if (current[0] == size - 1 && current[1] == size - 1) {
return current[2];
}
for (int[] direction : directions) {
int newX = current[0] + direction[0];
int newY = current[1] + direction[1];
if (cellIsValid(safetyGrid, newX, newY) && safetyGrid[newX][newY] != -1) {
maxSafetyQueue.add(new int[] {newX, newY, Math.min(current[2], safetyGrid[newX][newY])});
safetyGrid[newX][newY] = -1;
}
}
}
return -1;
}
// Validate grid cell position
private boolean cellIsValid(int[][] array, int xIndex, int yIndex) {
int bounds = array.length;
return xIndex >= 0 && yIndex >= 0 && xIndex < bounds && yIndex < bounds;
}
}
The given Java code implements a method to find the safest path in a grid by utilizing a breadth-first search (BFS) algorithm to compute a safety factor and then a priority queue to identify the path with the maximum safety.
- In the
calculateMaxSafetymethod, the input is a list of lists representing a matrix where each cell can be either0or1. Cells marked1are considered safe starting points. - The matrix size is determined, and a
safetyGridis created to store safety levels, which are initialized to-1except for safe cells (value of1in the original matrix) which are initialized to0. - A queue (
processingQueue) is used to implement BFS starting from all initial safe cells. For each cell processed, its adjacent cells (up, down, left, right) are checked—if they are valid and not visited, their safety level is updated, and they are added to the queue. - After populating the
safetyGridwith safety values through BFS, a priority queue (maxSafetyQueue) is employed to traverse the grid from the top-left to the bottom-right corner, attempting to maintain the highest possible safety level. At each step, the path with the current maximum safety value is chosen, and this value adjusts based on the minimum of the current and the new cell's safety level. - The traversal continues until the bottom-right corner of the grid is reached, at which point the path's safety level is returned.
- The
cellIsValidhelper method confirms whether a given cell lies within the bounds of the grid.
The solution effectively combines BFS for safety calculation and a priority-based greedy approach for pathfinding, ensuring optimal safety throughout the traversal from start to endpoint.
class Solution:
# Movement directions: right, left, down, up
movements = [(0, 1), (0, -1), (1, 0), (-1, 0)]
def findMaxSafeness(self, matrix: List[List[int]]) -> int:
size = len(matrix)
multi_level_bfs_queue = deque()
# Initialize matrix with distinctions for thieves and empty spaces
for row in range(size):
for col in range(size):
if matrix[row][col] == 1:
multi_level_bfs_queue.append((row, col))
matrix[row][col] = 0
else:
matrix[row][col] = -1
# Compute safeness recursively
while multi_level_bfs_queue:
current_level_size = len(multi_level_bfs_queue)
while current_level_size > 0:
current = multi_level_bfs_queue.popleft()
# Process all valid adjacent cells
for move in self.movements:
new_row, new_col = current[0] + move[0], current[1] + move[1]
current_value = matrix[current[0]][current[1]]
if self.cellWithinBounds(matrix, new_row, new_col) and matrix[new_row][new_col] == -1:
matrix[new_row][new_col] = current_value + 1
multi_level_bfs_queue.append((new_row, new_col))
current_level_size -= 1
# Implement max safeness path finder using PQ
priority_queue = [[-matrix[0][0], 0, 0]]
matrix[0][0] = -1
while priority_queue:
safeness, row, col = heapq.heappop(priority_queue)
if row == size - 1 and col == size - 1:
return -safeness
for move in self.movements:
newRow, newCol = row + move[0], col + move[1]
if self.cellWithinBounds(matrix, newRow, newCol) and matrix[newRow][newCol] != -1:
heapq.heappush(priority_queue, [-min(-safeness, matrix[newRow][newCol]), newRow, newCol])
matrix[newRow][newCol] = -1
return -1
def cellWithinBounds(self, matrix, row, col) -> bool:
return 0 <= row < len(matrix) and 0 <= col < len(matrix)
Implement a function findMaxSafeness in Python to determine the safest path through a grid represented by a matrix. Each cell in the grid can contain threats or be safe, and the goal is to traverse from the top left corner to the bottom right corner while maximizing the safeness of the path.
In this implementation, use the following approach:
- Analyze the matrix and convert it for processing, where cells with '1' are safe spaces and are initialized accordingly in a BFS queue while other cells are marked as 'not visited'.
- Use BFS to spread the safeness level from each '1' cell through all other cells they can influence, thus altering their initial state to a state representing their relative safety.
- To find the path, leverage a priority queue starting from the first cell and push the safeness levels of subsequent cells while making sure to not revisit cells.
- During the BFS process, adjust cells' values based on their maximum achievable safeness by comparison to their neighbors. Keep record of these values.
- Continue exploring available paths using the priority queue until reaching the target cell or exhausting all options.
- The function finally returns the maximum safeness encountered at the endpoint or returns -1 if no path is available.
Function Details:
- Use the deque for BFS queue operations to efficiently pop and append elements.
cellWithinBoundsensures movements stay within the grid limits.- Use heapq for the priority queue to always process the currently safest available path.
This method efficiently calculates the safest path by considering all potential paths' safeness levels and dynamically deciding the best route based on real-time calculations during execution.
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