Two Rats in a Maze

Given a 2D matrix maze[][], where some cells are blocked, some cells are locked (which the rats can unlock) and some cells are vacant. Given the initial position of two rats, find the minimum number of cells they need to unlock to escape the maze. A rat can escape the maze if it reaches outside the boundary of the maze. Blocked cells are represented by '*', locked cells are represented by '#', vacant cells are represented by '.', and the initial position of rats is represented by '$'. A path from the outside to that rat is guaranteed to exist for each rat.

Note: If a rat unlocks a cell, it remains unlocked. A rat cannot pass through blocked cells and can pass through locked cells after unlocking them. Add images to represent the problem.

Example:

Input: n = 5, m = 9,
inputMaze[5][10] = { ****#****
*..#.#..*
****.****
*$#.#.#$*
*********}
Output: 4

Input: n = 5, m = 11
inputMaze = {*#*********
*$*...*...*
*$*.*.*.*.*
*...*...*.*
*********.*}
Output: 0

Approach:

The idea is to first initializes distances from the outside boundary and rats' initial positions using breadth-first search (BFS). It then identifies and stores the positions of the locked cells. The BFS is performed for each rat separately, updating distances and considering the unlocking of doors. The final answer is calculated by adding the distances of both rats and the initial distances, adjusting for the unlocking of shared doors. The result represents the minimum number of cells that both rats need to unlock to escape the maze.

Steps:

• Initialization:
  • The maze is represented as a 2D grid (maze[][]).
  • The rats' initial positions are marked with '$', blocked cells with '*', locked cells with '#', and vacant cells with '.'.
• BFS for Initial Cells:
  • Initialize the distances from the outside boundary to the initial cells with money ('$') and doors ('#').
  • Perform a breadth-first search (BFS) from the border cells with money ('$') and doors ('#').
  • Update the distances as the BFS progresses.
• Identification of Rats and Locked Cells:
  • Identify the positions of the two rats and collect the positions of unlocked doors ('#').
• DFS and BFS for Rats and Unlocked Doors:
  • Perform depth-first search (DFS) and breadth-first search (BFS) to calculate the distances from the outside boundary to each rat and the distances from each rat to every unlocked door.
  • Update the distances during the searches.
• Calculation of Minimum Cells to Unlock:
  • Calculate the minimum number of cells the rats need to unlock to escape by considering their individual distances and the initial distances from the outside boundary.
  • Iterate through the positions of unlocked doors and update the answer based on the sum of distances from both rats to the door and subtracting the initial distances (accounting for the fact that each door is unlocked by both rats).
• Final Answer:
  • The final answer represents the minimum number of cells the two rats need to unlock collectively to successfully escape the maze.

Below is the implementation of the above approach:

4

Time complexity: O(N * M), where N is the number of rows and M is the number of columns in the maze. The primary factor contributing to this complexity is the breadth-first search (BFS) operation, which explores the maze cells.

Auxiliary Space : O(N * M), where N is the number of rows and M is the number of columns in the maze.