Maze with N doors and 1 Key

Moderate

0/80

Average time to solve is 15m

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12 upvotes

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Problem statement

You are given an 'N * N' 'MAZE' where some cells have a door while some do not and a key that can be used only once to open a door.

You need to find if there exists a path from the top-left cell to the bottom right cell of the maze provided only downward and rightward movements are allowed.

Note:

1. You have only one key. And a key once used is exhausted and no more available with you during the journey through that path in a maze.

2. A cell with value 1, means the door or path is closed. And you have to spend a key to open the door/ reach that cell.

3. A cell with value 0, means that the cell is free to move / door is always open.

4. Top left cell in the maze and bottom-right cell in the maze may also have a door.

5. Downwards movement: From cell (i, j) to (i, j+1).

6. Rightwards movement: From cell (i, j) to (i+1, j).

Detailed explanation ( Input/output format, Notes, Images )

Input Format:

The first line of the input contains an integer 'T' denoting the number of test cases. 
Then 'T' test cases follow.

The first line of each test case contains the side length 'N' of the square binary maze.

Then 'N' lines follow.
Each line contains 'N' space-separated integers 1 or 0 denoting whether the cell has a door or not.

Output Format:

For each test case, print in a separate line “YES” if the bottom right corner is reachable, else print “NO”.

Note:

You do not need to print anything, it has already been taken care of. Just implement the given function.

Constraints:

1 <= 'T' <= 100
1 <= 'N' <= 100
'MAZE[i][j]' = {0, 1}

Time Limit: 1 sec

Sample Input 1:

Sample Output 1:

YES

Explanation for Sample Input 1:

There are 3 paths possible; two paths have been shown in the below diagram. Note that we are using our only key at the cell (2,3) (1-based indexing).

Sample Input 2:

Sample Output 2:

NO

Hint 1

Hint 2

Depth-First Search, Recursion, BackTracking.

Approaches (2)

Approach 1

Approach 2

Recursion

This problem can be solved using Recursion.
If the current position in the maze (x, y) is set to 0, that means no key is required to open the current door. Check if it is destination return true. Else, move to next possible positions (right and bottom cells, if they lie in the maze).
If the current cell in the maze has a door(i.e the value of the cell is 1), then the key is needed to open the door. If the key has not been used previously in this path, use the key, i.e. set key = false. Check if its destination, else move to next possible positions (right and bottom cells, if they lie in the maze).
If any path reaches destination / bottom-right cell of the maze, print “YES”. Else print “NO”.

Time Complexity

O(2^N), where ‘N’ is the side length of the maze.

As we have 2 choices on each level of our recursion tree which will take O(2^N) to check all paths from top leftmost cell to bottom right cell.

Space Complexity

O(N*N), where ‘N’ is the side length of the maze.

The space required is proportional to the maximum depth of the recursion tree. So the space complexity is O(N*N).

(100% EXP penalty)

Maze with N doors and 1 Key

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