You are given an m x n grid where each cell can have one of three values:

  • 0 representing an empty cell,
  • 1 representing a fresh orange, or
  • 2 representing a rotten orange.

Every minute, any fresh orange that is 4-directionally adjacent to a rotten orange becomes rotten.

Return the minimum number of minutes that must elapse until no cell has a fresh orange. If this is impossible, return -1.

Examples

Example 1:

Input: grid = [[2,1,1],[1,1,0],[0,1,1]]
Output: 4

Example 2:

Input: grid = [[2,1,1],[0,1,1],[1,0,1]]
Output: -1
Explanation: The orange in the bottom left corner (row 2, column 0) is never rotten, because rotting only happens 4-directionally.

Example 3:

Input: grid = [[0,2]]
Output: 0
Explanation: Since there are no fresh oranges at minute 0, the answer is just 0.

Constraints

  • m == grid.length
  • n == grid[i].length
  • 1 <= m, n <= 10
  • grid[i][j] is 0, 1, or 2.

Thinking Process

  1. Multi-source BFS: Start from all rotten oranges simultaneously
  • 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 rotten oranges simultaneously. A special marker {-1, -1} is used to separate levels (minutes).

from collections import deque

class Solution:
    def orangesRotting(self, grid):
        cache = deque()
        freshOranges = 0

        ROWS = len(grid)
        COLS = len(grid[0])

        # Find all rotten oranges and count fresh oranges
        for r in range(ROWS):
            for c in range(COLS):
                if grid[r][c] == 2:
                    cache.append((r, c))
                elif grid[r][c] == 1:
                    freshOranges += 1

        # Add level separator
        cache.append((-1, -1))

        minutesElapsed = -1

        dirs = [(-1, 0), (1, 0), (0, 1), (0, -1)]

        while cache:
            row, col = cache.popleft()

            if row == -1:
                # Level separator encountered
                minutesElapsed += 1

                if cache:
                    # Add separator for next level
                    cache.append((-1, -1))
            else:
                # Process current rotten orange
                for d in dirs:
                    nr = row + d[0]
                    nc = col + d[1]

                    if (0 <= nr < ROWS and 0 <= nc < COLS and
                        grid[nr][nc] == 1):

                        grid[nr][nc] = 2  # Mark as rotten
                        freshOranges -= 1
                        cache.append((nr, nc))

        return minutesElapsed if freshOranges == 0 else -1

Solution Explanation

Approach: Queue BFS (this problem)

Key idea: 1. Multi-source BFS: Start from all rotten oranges simultaneously

How the code works:

  1. Multi-source BFS: Start from all rotten oranges simultaneously
    • 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 grid = [[2,1,1],[1,1,0],[0,1,1]], expected output 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
Initial scan 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 rotten oranges (value 2) and add them to queue
    • Count all fresh oranges (value 1)
    • Add level separator {-1, -1} after initial rotten oranges
  2. BFS Processing:
    • When encountering {-1, -1}, increment minutes and add separator for next level
    • For each rotten orange, check 4 neighbors
    • If neighbor is fresh, mark it as rotten, decrement fresh count, and add to queue
  3. Result:
    • If all fresh oranges are rotten (freshOranges == 0), return minutes elapsed
    • Otherwise, return -1 (impossible to rot all oranges)

Key Insight

The {-1, -1} marker acts as a level separator:

  • All oranges at the same level (same minute) are processed together
  • When we encounter the marker, we know we’ve finished one minute
  • This allows us to track time without maintaining a separate distance/time array

    Example Walkthrough

Input: grid = [[2,1,1],[1,1,0],[0,1,1]]

Solution 1 (Marker-based):

Initial:
  Rotten: (0,0)
  Fresh: 5 oranges
  Queue: [(0,0), (-1,-1)]

Minute 0:
  Process (0,0): Rot (0,1) and (1,0)
  Queue: [(-1,-1), (0,1), (1,0)]
  Fresh: 3

Minute 1:
  Process (0,1): Rot (0,2)
  Process (1,0): Rot (1,1)
  Queue: [(-1,-1), (0,2), (1,1)]
  Fresh: 1

Minute 2:
  Process (0,2): No new rots
  Process (1,1): Rot (2,1)
  Queue: [(-1,-1), (2,1)]
  Fresh: 0

Minute 3:
  Process (2,1): Rot (2,2)
  Queue: [(-1,-1), (2,2)]
  Fresh: 0

Minute 4:
  Process (2,2): No new rots
  Queue: [(-1,-1)]
  Fresh: 0

Result: 4 minutes

Complexity

| Operation | Time | Space | |———–|——|——-| | Initial scan | O(m×n) | O(1) | | BFS traversal | O(m×n) | O(m×n) | | Overall | O(m×n) | O(m×n) |

Common Mistakes

  1. No fresh oranges: [[0,2]] → return 0
  2. Impossible to rot all: Isolated fresh orange → return -1
  3. All rotten initially: [[2,2]] → return 0
  4. All fresh initially: No rotten oranges → return -1

  5. Empty grid: Not possible per constraints

  6. Not counting initial fresh oranges: Must count before BFS starts
  7. Wrong level tracking: Forgetting to increment minutes at right time
  8. Boundary checks: Not checking array bounds before accessing
  9. Visited tracking: Not marking as rotten immediately (could process same cell twice)
  10. Return value: Returning minutes when freshOranges > 0 (should return -1)

Optimization Tips

  1. Early termination: Can break early if freshOranges == 0 during BFS
  2. Space optimization: Solution 1 uses no extra space beyond queue
  3. Level tracking: Marker approach is elegant but level-size approach is clearer

Pattern Recognition

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

1. Find all starting points (sources)
2. Add all sources to queue
3. Process level by level
4. Track time/distance from sources
5. Check if all targets are reached

Similar problems:

  • Walls and Gates
  • Shortest Distance from All Buildings
  • 01 Matrix
  • As Far from Land as Possible

Real-World Applications

  1. Disease Spread: Model how disease spreads from multiple sources
  2. Network Broadcasting: Broadcast message from multiple nodes
  3. Fire Spread: Simulate fire spreading from multiple ignition points
  4. Virus Propagation: Model computer virus spreading in network
  5. Information Diffusion: Track information spread in social networks

    References

Key Takeaways

  1. Multi-source BFS: Start from all rotten oranges simultaneously
  2. Level separation: Need to track time/levels to know when all oranges at current level are processed
  3. Fresh count tracking: Decrement count when rotting, check if zero at end
  4. Grid modification: Mark as rotten immediately to avoid reprocessing