This problem asks to find a path in a grid that maximizes safeness from thieves. The approach combines binary search and BFS to efficiently calculate the maximum safeness factor. We search over the possible safeness values using binary search, checking each potential value with BFS to verify if a safe path exists.
Problem Statement
You are given a 0-indexed 2D matrix grid of size n x n, where each cell is either 0 (empty) or 1 (contains a thief). Starting from the top-left corner (0, 0), your task is to find the safest path to the bottom-right corner (n-1, n-1). In each move, you can travel to any adjacent cell, including those containing thieves. The objective is to determine the maximum safeness factor for any valid path.
The safeness factor of a path is defined as the minimum Manhattan distance from any cell in the path to any thief. You are to compute the maximum safeness factor achievable for a valid path from the top-left to the bottom-right corner of the grid, while considering obstacles such as thieves.
Examples
Example 1
Input: grid = [[1,0,0],[0,0,0],[0,0,1]]
Output: 0
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
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
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 <= 400
- grid[i].length == n
- grid[i][j] is either 0 or 1.
- There is at least one thief in the grid.
Solution Approach
Binary Search Over Safeness Factor
The core of the solution uses binary search over possible safeness factors, from 0 to n. This approach works because we can narrow down the range of safeness by testing each candidate value, reducing the complexity of brute-forcing all possible paths.
Breadth-First Search (BFS) to Validate Paths
For each candidate safeness factor from the binary search, we use BFS to check whether there exists a valid path where all points have at least that safeness factor. BFS ensures that we efficiently explore the grid from the top-left to bottom-right corners while adhering to the safeness constraint.
Combining Binary Search and BFS
By combining binary search with BFS, we balance efficiency and correctness. Binary search reduces the number of possible safeness factors to check, while BFS ensures that we find a valid path for each safeness level tested. This combination optimizes the overall solution.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n^2 \cdot \log (n)) |
| Space | O(n^2) |
The time complexity is O(n^2 * log(n)) because for each of the O(log(n)) iterations of binary search, we perform an O(n^2) BFS to check if a path with the given safeness factor exists. The space complexity is O(n^2) due to the grid and BFS queue.
What Interviewers Usually Probe
- Candidate effectively applies binary search over a range of potential safeness factors.
- Candidate demonstrates clear understanding of BFS for pathfinding in a grid with constraints.
- Candidate explains the trade-offs in combining binary search and BFS to optimize the solution.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to account for the thieves in the grid when calculating safeness factor.
- Misunderstanding the binary search range, which may lead to incorrect results.
- Not efficiently handling the BFS queue, leading to performance issues for larger grids.
Follow-up variants
- Handling grids with varying thief positions and grid sizes.
- Modifying the problem to account for multiple thieves instead of a single thief.
- Changing the grid shape, such as making it non-square, and adjusting the approach.
How GhostInterview Helps
- GhostInterview helps by providing clear insights into solving grid-based pathfinding problems using advanced techniques like BFS and binary search.
- GhostInterview guides users through the complexities of applying binary search over a valid solution space for optimization.
- GhostInterview helps candidates avoid common pitfalls such as inefficient BFS or incorrect safeness calculations.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
How do I approach the 'Find the Safest Path in a Grid' problem?
The problem can be solved efficiently using a combination of binary search over safeness factors and BFS to validate paths at each safeness level.
What is the primary pattern in the 'Find the Safest Path in a Grid' problem?
The primary pattern is binary search over the valid answer space, combined with BFS for path validation at each candidate safeness factor.
What is the time complexity of the solution?
The time complexity is O(n^2 * log(n)), where n is the grid size. The binary search performs O(log(n)) checks, each requiring an O(n^2) BFS.
Can BFS alone solve the problem?
BFS alone isn't enough because we need to search over different safeness factors, which is why binary search is needed to optimize the solution.
What are the key trade-offs when using binary search and BFS together?
The main trade-off is balancing the efficiency of binary search with the correctness of BFS. While binary search reduces the number of possible safeness levels to check, BFS ensures each path is validated against the safeness constraint.
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