# Rat in a Maze | Backtracking-2

We have discussed Backtracking and Knight’s tour problem in Set 1. Let us discuss Rat in a Maze as another example problem that can be solved using Backtracking.

A Maze is given as N*N binary matrix of blocks where source block is the upper left most block i.e., maze and destination block is lower rightmost block i.e., maze[N-1][N-1]. A rat starts from source and has to reach the destination. The rat can move only in two directions: forward and down.
In the maze matrix, 0 means the block is a dead end and 1 means the block can be used in the path from source to destination. Note that this is a simple version of the typical Maze problem. For example, a more complex version can be that the rat can move in 4 directions and a more complex version can be with a limited number of moves.

Following is an example maze.

` Gray blocks are dead ends (value = 0). ` Following is binary matrix representation of the above maze.

```                {1, 0, 0, 0}
{1, 1, 0, 1}
{0, 1, 0, 0}
{1, 1, 1, 1}
```

Following is a maze with highlighted solution path. Following is the solution matrix (output of program) for the above input matrx.

```                {1, 0, 0, 0}
{1, 1, 0, 0}
{0, 1, 0, 0}
{0, 1, 1, 1}
All enteries in solution path are marked as 1.
```

Naive Algorithm
The Naive Algorithm is to generate all paths from source to destination and one by one check if the generated path satisfies the constraints.

```while there are untried paths
{
generate the next path
if this path has all blocks as 1
{
print this path;
}
}```

Backtracking Algorithm

```If destination is reached
print the solution matrix
Else
a) Mark current cell in solution matrix as 1.
b) Move forward in the horizontal direction and recursively check if this
c) If the move chosen in the above step doesn't lead to a solution
then move down and check if this move leads to a solution.
d) If none of the above solutions works then unmark this cell as 0
(BACKTRACK) and return false.
```

Implementation of Backtracking solution

## C/C++

 `/* C/C++ program to solve Rat in a Maze problem using ` `   ``backtracking */` `#include ` ` `  `// Maze size ` `#define N 4  ` ` `  `bool` `solveMazeUtil(``int` `maze[N][N], ``int` `x, ``int` `y, ``int` `sol[N][N]); ` ` `  `/* A utility function to print solution matrix sol[N][N] */` `void` `printSolution(``int` `sol[N][N]) ` `{ ` `    ``for` `(``int` `i = 0; i < N; i++) ` `    ``{ ` `        ``for` `(``int` `j = 0; j < N; j++) ` `            ``printf``(``" %d "``, sol[i][j]); ` `        ``printf``(````" "````); ` `    ``} ` `} ` ` `  `/* A utility function to check if x,y is valid index for N*N maze */` `bool` `isSafe(``int` `maze[N][N], ``int` `x, ``int` `y) ` `{ ` `    ``// if (x,y outside maze) return false ` `    ``if``(x >= 0 && x < N && y >= 0 && y < N && maze[x][y] == 1) ` `        ``return` `true``; ` ` `  `    ``return` `false``; ` `} ` ` `  `/* This function solves the Maze problem using Backtracking.  It mainly ` `   ``uses solveMazeUtil() to solve the problem. It returns false if no  ` `   ``path is possible, otherwise return true and prints the path in the ` `   ``form of 1s. Please note that there may be more than one solutions,  ` `   ``this function prints one of the feasible solutions.*/` `bool` `solveMaze(``int` `maze[N][N]) ` `{ ` `    ``int` `sol[N][N] = { {0, 0, 0, 0}, ` `        ``{0, 0, 0, 0}, ` `        ``{0, 0, 0, 0}, ` `        ``{0, 0, 0, 0} ` `    ``}; ` ` `  `    ``if``(solveMazeUtil(maze, 0, 0, sol) == ``false``) ` `    ``{ ` `        ``printf``(``"Solution doesn't exist"``); ` `        ``return` `false``; ` `    ``} ` ` `  `    ``printSolution(sol); ` `    ``return` `true``; ` `} ` ` `  `/* A recursive utility function to solve Maze problem */` `bool` `solveMazeUtil(``int` `maze[N][N], ``int` `x, ``int` `y, ``int` `sol[N][N]) ` `{ ` `    ``// if (x,y is goal) return true ` `    ``if``(x == N-1 && y == N-1) ` `    ``{ ` `        ``sol[x][y] = 1; ` `        ``return` `true``; ` `    ``} ` ` `  `    ``// Check if maze[x][y] is valid ` `    ``if``(isSafe(maze, x, y) == ``true``) ` `    ``{ ` `        ``// mark x,y as part of solution path ` `        ``sol[x][y] = 1; ` ` `  `        ``/* Move forward in x direction */` `        ``if` `(solveMazeUtil(maze, x+1, y, sol) == ``true``) ` `            ``return` `true``; ` ` `  `        ``/* If moving in x direction doesn't give solution then ` `           ``Move down in y direction  */` `        ``if` `(solveMazeUtil(maze, x, y+1, sol) == ``true``) ` `            ``return` `true``; ` ` `  `        ``/* If none of the above movements work then BACKTRACK:  ` `            ``unmark x,y as part of solution path */` `        ``sol[x][y] = 0; ` `        ``return` `false``; ` `    ``}    ` ` `  `    ``return` `false``; ` `} ` ` `  `// driver program to test above function ` `int` `main() ` `{ ` `    ``int` `maze[N][N]  =  { {1, 0, 0, 0}, ` `        ``{1, 1, 0, 1}, ` `        ``{0, 1, 0, 0}, ` `        ``{1, 1, 1, 1} ` `    ``}; ` ` `  `    ``solveMaze(maze); ` `    ``return` `0; ` `} `

## Java

 `/* Java program to solve Rat in a Maze problem using ` `   ``backtracking */` ` `  `public` `class` `RatMaze ` `{ ` `    ``final` `int` `N = ``4``; ` ` `  `    ``/* A utility function to print solution matrix ` `       ``sol[N][N] */` `    ``void` `printSolution(``int` `sol[][]) ` `    ``{ ` `        ``for` `(``int` `i = ``0``; i < N; i++) ` `        ``{ ` `            ``for` `(``int` `j = ``0``; j < N; j++) ` `                ``System.out.print(``" "` `+ sol[i][j] + ` `                                 ``" "``); ` `            ``System.out.println(); ` `        ``} ` `    ``} ` ` `  `    ``/* A utility function to check if x,y is valid ` `        ``index for N*N maze */` `    ``boolean` `isSafe(``int` `maze[][], ``int` `x, ``int` `y) ` `    ``{ ` `        ``// if (x,y outside maze) return false ` `        ``return` `(x >= ``0` `&& x < N && y >= ``0` `&& ` `                ``y < N && maze[x][y] == ``1``); ` `    ``} ` ` `  `    ``/* This function solves the Maze problem using ` `       ``Backtracking. It mainly uses solveMazeUtil() ` `       ``to solve the problem. It returns false if no ` `       ``path is possible, otherwise return true and ` `       ``prints the path in the form of 1s. Please note ` `       ``that there may be more than one solutions, this ` `       ``function prints one of the feasible solutions.*/` `    ``boolean` `solveMaze(``int` `maze[][]) ` `    ``{ ` `        ``int` `sol[][] = {{``0``, ``0``, ``0``, ``0``}, ` `            ``{``0``, ``0``, ``0``, ``0``}, ` `            ``{``0``, ``0``, ``0``, ``0``}, ` `            ``{``0``, ``0``, ``0``, ``0``} ` `        ``}; ` ` `  `        ``if` `(solveMazeUtil(maze, ``0``, ``0``, sol) == ``false``) ` `        ``{ ` `            ``System.out.print(``"Solution doesn't exist"``); ` `            ``return` `false``; ` `        ``} ` ` `  `        ``printSolution(sol); ` `        ``return` `true``; ` `    ``} ` ` `  `    ``/* A recursive utility function to solve Maze ` `       ``problem */` `    ``boolean` `solveMazeUtil(``int` `maze[][], ``int` `x, ``int` `y, ` `                          ``int` `sol[][]) ` `    ``{ ` `        ``// if (x,y is goal) return true ` `        ``if` `(x == N - ``1` `&& y == N - ``1``) ` `        ``{ ` `            ``sol[x][y] = ``1``; ` `            ``return` `true``; ` `        ``} ` ` `  `        ``// Check if maze[x][y] is valid ` `        ``if` `(isSafe(maze, x, y) == ``true``) ` `        ``{ ` `            ``// mark x,y as part of solution path ` `            ``sol[x][y] = ``1``; ` ` `  `            ``/* Move forward in x direction */` `            ``if` `(solveMazeUtil(maze, x + ``1``, y, sol)) ` `                ``return` `true``; ` ` `  `            ``/* If moving in x direction doesn't give ` `               ``solution then  Move down in y direction */` `            ``if` `(solveMazeUtil(maze, x, y + ``1``, sol)) ` `                ``return` `true``; ` ` `  `            ``/* If none of the above movements works then ` `               ``BACKTRACK: unmark x,y as part of solution ` `               ``path */` `            ``sol[x][y] = ``0``; ` `            ``return` `false``; ` `        ``} ` ` `  `        ``return` `false``; ` `    ``} ` ` `  `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``RatMaze rat = ``new` `RatMaze(); ` `        ``int` `maze[][] = {{``1``, ``0``, ``0``, ``0``}, ` `            ``{``1``, ``1``, ``0``, ``1``}, ` `            ``{``0``, ``1``, ``0``, ``0``}, ` `            ``{``1``, ``1``, ``1``, ``1``} ` `        ``}; ` `        ``rat.solveMaze(maze); ` `    ``} ` `} ` `// This code is contributed by Abhishek Shankhadhar `

## Python3

 `# Python3 program to solve Rat in a Maze  ` `# problem using backracking  ` ` `  `# Maze size ` `N ``=` `4` ` `  `# A utility function to print solution matrix sol ` `def` `printSolution( sol ): ` `     `  `    ``for` `i ``in` `sol: ` `        ``for` `j ``in` `i: ` `            ``print``(``str``(j) ``+` `" "``, end``=``"") ` `        ``print``("") ` ` `  `# A utility function to check if x,y is valid ` `# index for N*N Maze ` `def` `isSafe( maze, x, y ): ` `     `  `    ``if` `x >``=` `0` `and` `x < N ``and` `y >``=` `0` `and` `y < N ``and` `maze[x][y] ``=``=` `1``: ` `        ``return` `True` `     `  `    ``return` `False` ` `  `""" This function solves the Maze problem using Backtracking.  ` `    ``It mainly uses solveMazeUtil() to solve the problem. It  ` `    ``returns false if no path is possible, otherwise return  ` `    ``true and prints the path in the form of 1s. Please note ` `    ``that there may be more than one solutions, this function ` `    ``prints one of the feasable solutions. """` `def` `solveMaze( maze ): ` `     `  `    ``# Creating a 4 * 4 2-D list ` `    ``sol ``=` `[ [ ``0` `for` `j ``in` `range``(``4``) ] ``for` `i ``in` `range``(``4``) ] ` `     `  `    ``if` `solveMazeUtil(maze, ``0``, ``0``, sol) ``=``=` `False``: ` `        ``print``(``"Solution doesn't exist"``); ` `        ``return` `False` `     `  `    ``printSolution(sol) ` `    ``return` `True` `     `  `# A recursive utility function to solve Maze problem ` `def` `solveMazeUtil(maze, x, y, sol): ` `     `  `    ``#if (x,y is goal) return True ` `    ``if` `x ``=``=` `N ``-` `1` `and` `y ``=``=` `N ``-` `1``: ` `        ``sol[x][y] ``=` `1` `        ``return` `True` `         `  `    ``# Check if maze[x][y] is valid ` `    ``if` `isSafe(maze, x, y) ``=``=` `True``: ` `        ``# mark x, y as part of solution path ` `        ``sol[x][y] ``=` `1` `         `  `        ``# Move forward in x direction ` `        ``if` `solveMazeUtil(maze, x ``+` `1``, y, sol) ``=``=` `True``: ` `            ``return` `True` `             `  `        ``# If moving in x direction doesn't give solution  ` `        ``# then Move down in y direction ` `        ``if` `solveMazeUtil(maze, x, y ``+` `1``, sol) ``=``=` `True``: ` `            ``return` `True` `         `  `        ``# If none of the above movements work then  ` `        ``# BACKTRACK: unmark x,y as part of solution path ` `        ``sol[x][y] ``=` `0` `        ``return` `False` ` `  `# Driver program to test above function ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``# Initialising the maze ` `    ``maze ``=` `[ [``1``, ``0``, ``0``, ``0``], ` `             ``[``1``, ``1``, ``0``, ``1``], ` `             ``[``0``, ``1``, ``0``, ``0``], ` `             ``[``1``, ``1``, ``1``, ``1``] ] ` `              `  `    ``solveMaze(maze) ` ` `  `# This code is contributed by Shiv Shankar `

Output: The 1 values show the path for rat

``` 1  0  0  0
1  1  0  0
0  1  0  0
0  1  1  1 ```

Below is an extended version of this problem.Count number of ways to reach destination in a Maze