You are given an m x n grid rooms initialized with these three possible values:

  • -1 A wall or an obstacle.
  • 0 A gate.
  • INF Infinity means an empty room. We use the value 2^31 - 1 = 2147483647 to represent INF as you may assume that the distance to a gate is less than 2147483647.

Fill each empty room with the distance to its nearest gate. If it is impossible to reach a gate, it should remain INF.

Examples

Example 1:

Input: rooms = [[2147483647,-1,0,2147483647],[2147483647,2147483647,2147483647,-1],[2147483647,-1,2147483647,-1],[0,-1,2147483647,2147483647]]
Output: [[3,-1,0,1],[2,2,1,-1],[1,-1,2,-1],[0,-1,3,4]]

Example 2:

Input: rooms = [[-1]]
Output: [[-1]]

Example 3:

Input: rooms = [[2147483647]]
Output: [[2147483647]]

Constraints

  • m == rooms.length
  • n == rooms[i].length
  • 1 <= m, n <= 250
  • rooms[i][j] is -1, 0, or 2^31 - 1.

Thinking Process

  1. Multi-source BFS: Start from all gates simultaneously for efficiency
  • BFS visits nodes in non-decreasing distance from the source.
  • Queue guarantees shortest path in unweighted graphs.
  • Process level by level when counting layers or distances.
Graph BFS layers S a b t BFS: expand by layers (queue)

Common Approaches

Typical techniques for this pattern:

Approach Time Space Notes
Queue BFS (this problem) O(n) O(n) Shortest path in unweighted graphs
Multi-source BFS O(n) O(n) Start from all sources simultaneously
0-1 BFS / deque O(n) O(n) Weights 0 or 1
Level-order BFS O(n) O(w) Process by depth/layer

Solution

Time Complexity: O(m × n) - Each cell is visited at most once
Space Complexity: O(m × n) - For the queue

This solution uses BFS starting from all gates simultaneously. Since BFS guarantees shortest path in unweighted graphs, each empty room will be filled with the distance to its nearest gate.

class Solution:
EMPTY = INT_MAX
GATE = 0
dirs[4][2] = ::1, 0, :-1, 0, :0, 1, :0, -1
def wallsAndGates(self, rooms):
    ROWS = len(rooms), COLS = rooms[0].__len__()
    if(ROWS == 0  or  COLS == 0) return
    deque[pair<int, int>> q
    # Add all gates to queue (multi-source BFS)
    for(row = 0 row < ROWS row += 1) :
    for(col = 0 col < COLS col += 1) :
    if rooms[row][col] == GATE:
        q.push(:row, col)
# BFS from all gates simultaneously
while not not q:
    [row, col] = q[0]
    q.pop()
    for dir in dirs:
        nrow = row + dir[0]
        ncol = col + dir[1]
        # Skip if out of bounds, wall, or already visited (not EMPTY)
        if(nrow < 0  or  nrow >= ROWS  or  ncol < 0  or  ncol >= COLS  or
        rooms[nrow][ncol] != EMPTY) :
        continue
    # Update distance and add to queue
    rooms[nrow][ncol] = rooms[row][col] + 1
    q.push(:nrow, ncol)

Solution Explanation

Approach: Queue BFS (this problem)

Key idea: 1. Multi-source BFS: Start from all gates simultaneously for efficiency

How the code works:

  1. Multi-source BFS: Start from all gates simultaneously for efficiency
    • BFS visits nodes in non-decreasing distance from the source.
    • Queue guarantees shortest path in unweighted graphs.
    • Process level by level when counting layers or distances.

Walkthrough — input rooms = [[2147483647,-1,0,2147483647],[2147483647,2147483647,2147483647,-1],[2147483647,-1,2147483647,-1],[0,-1,2147483647,2147483647]], expected output [[3,-1,0,1],[2,2,1,-1],[1,-1,2,-1],[0,-1,3,4]]:

  1. Initialize variables from the problem setup.
  2. Apply the main loop / recursion until the condition is met.
  3. Confirm the result matches the expected output.
Operation Time Space
Find all gates O(m×n) O(1)
BFS traversal O(m×n) O(m×n)
Overall O(m×n) O(m×n)

How Solution 1 Works

  1. Initialization:
    • Find all gates (value 0) and add them to the queue
    • Gates are the starting points for BFS
  2. Multi-Source BFS:
    • Process all gates simultaneously
    • For each gate’s neighbor:
      • If it’s an empty room (EMPTY), update its distance
      • Distance = current cell’s distance + 1
      • Add to queue for further exploration
  3. Key Insight:
    • BFS guarantees shortest path in unweighted graphs
    • By starting from all gates, each empty room gets the distance to its nearest gate
    • Walls (-1) are skipped, empty rooms are updated in-place

Why Multi-Source BFS?

  • Single-source BFS would require running BFS from each gate separately → O(k × m × n) where k is number of gates
  • Multi-source BFS processes all gates together → O(m × n)
  • Each cell is visited once, and the first visit (from nearest gate) sets the correct distance

    Example Walkthrough

Input:

rooms = [
  [INF, -1,  0,  INF],
  [INF, INF, INF, -1 ],
  [INF, -1,  INF, -1 ],
  [0,   -1,  INF, INF]
]

Solution 1 (Multi-Source BFS):

Step 1: Find all gates
  Gates at: (0,2) and (3,0)
  Queue: [(0,2), (3,0)]

Step 2: BFS from gates
  Process (0,2):
    Neighbors: (1,2), (-1,2), (0,1), (0,3)
    Valid: (1,2), (0,3)
    Update: rooms[1][2] = 1, rooms[0][3] = 1
    Queue: [(3,0), (1,2), (0,3)]
  
  Process (3,0):
    Neighbors: (4,0), (2,0), (3,1), (3,-1)
    Valid: (2,0)
    Update: rooms[2][0] = 1
    Queue: [(1,2), (0,3), (2,0)]
  
  Process (1,2):
    Neighbors: (2,2), (0,2), (1,1), (1,3)
    Valid: (2,2), (1,1)
    Update: rooms[2][2] = 2, rooms[1][1] = 2
    Queue: [(0,3), (2,0), (2,2), (1,1)]
  
  Continue until queue is empty...

Final Result:
[
  [3, -1,  0,  1],
  [2,  2,  1, -1],
  [1, -1,  2, -1],
  [0, -1,  3,  4]
]

Complexity

| Operation | Time | Space | |———–|——|——-| | Find all gates | O(m×n) | O(1) | | BFS traversal | O(m×n) | O(m×n) | | Overall | O(m×n) | O(m×n) |

Common Mistakes

  1. No gates: All rooms remain INF
  2. No empty rooms: Only walls and gates
  3. Single cell: [[-1]] or [[0]] or [[INF]]
  4. All gates: Every cell is a gate

  5. Isolated rooms: Rooms that cannot reach any gate remain INF

  6. Single-source BFS: Running BFS from each gate separately (inefficient)
  7. Not checking bounds: Forgetting to check array bounds before accessing
  8. Wrong condition: Using rooms[nrow][ncol] == EMPTY instead of != EMPTY for visited check
  9. Distance calculation: Forgetting to add 1 when updating distance
  10. Modifying input: Not handling the case where input is empty

Optimization Tips

  1. Early termination: Not applicable here (need to fill all rooms)
  2. Space optimization: Solution 1 uses grid itself, no extra distance array
  3. Direction array: Using constant array for directions is cleaner than nested loops

Pattern Recognition

This problem demonstrates the “Multi-Source BFS” pattern:

1. Identify all starting points (gates)
2. Add all sources to queue
3. Process level by level (BFS)
4. Update distances in-place
5. Skip obstacles (walls)

Similar problems:

  • Rotting Oranges
  • 01 Matrix
  • Shortest Distance from All Buildings
  • As Far from Land as Possible

Real-World Applications

  1. Navigation Systems: Find nearest exit/entrance in a building
  2. Game Development: Pathfinding from multiple spawn points
  3. Network Routing: Find nearest gateway/router
  4. Emergency Evacuation: Find nearest exit in emergency scenarios
  5. Facility Location: Find optimal locations based on multiple sources

    References

Key Takeaways

  1. Multi-source BFS: Start from all gates simultaneously for efficiency
  2. In-place update: Use the grid itself to track distances (no extra space needed)
  3. BFS guarantees shortest path: First visit to a cell gives the shortest distance
  4. Walls are obstacles: Skip walls (-1) during traversal
  5. Empty check: Only process cells that are EMPTY (not yet visited)