# Rat in a Maze | Backtracking-2

• Difficulty Level : Medium
• Last Updated : 26 May, 2022

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[0][0] 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 a 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 matrix.

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

Backtracking Algorithm: Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally. Solving one piece at a time, and removing those solutions that fail to satisfy the constraints of the problem at any point of time (by time, here, is referred to the time elapsed till reaching any level of the search tree) is the process of backtracking.

Approach: Form a recursive function, which will follow a path and check if the path reaches the destination or not. If the path does not reach the destination then backtrack and try other paths.

Algorithm:

1. Create a solution matrix, initially filled with 0’s.
2. Create a recursive function, which takes initial matrix, output matrix and position of rat (i, j).
3. if the position is out of the matrix or the position is not valid then return.
4. Mark the position output[i][j] as 1 and check if the current position is destination or not. If destination is reached print the output matrix and return.
5. Recursively call for position (i+1, j) and (i, j+1).
6. Unmark position (i, j), i.e output[i][j] = 0.

## C++

 `// C++ program to solve Rat in a Maze problem using` `// backtracking` `#include ` `using` `namespace` `std;` `// 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++)` `            ``cout<<``" "``<= 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``) {` `        ``cout<<``"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 && maze[x][y] == 1) {` `        ``sol[x][y] = 1;` `        ``return` `true``;` `    ``}` `    ``// Check if maze[x][y] is valid` `    ``if` `(isSafe(maze, x, y) == ``true``) {` `        ``// Check if the current block is already part of` `        ``// solution path.` `        ``if` `(sol[x][y] == 1)` `            ``return` `false``;` `        ``// 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;` `}`   `// This code is contributed by Aditya Kumar (adityakumar129)`

## C

 `// C++ program to solve Rat in a Maze problem using` `// backtracking` `#include ` `#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``(``"\n"``);` `    ``}` `}`   `// 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 && maze[x][y] == 1) {` `        ``sol[x][y] = 1;` `        ``return` `true``;` `    ``}` `    ``// Check if maze[x][y] is valid` `    ``if` `(isSafe(maze, x, y) == ``true``) {` `        ``// Check if the current block is already part of` `        ``// solution path.` `        ``if` `(sol[x][y] == 1)` `            ``return` `false``;` `        ``// 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;` `}`   `// This code is contributed by Aditya Kumar (adityakumar129)`

## Java

 `/* Java program to solve Rat in` ` ``a Maze problem using backtracking */`   `public` `class` `RatMaze {`   `    ``// Size of the maze` `    ``static` `int` `N;`   `    ``/* 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[][] = ``new` `int``[N][N];`   `        ``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` `            ``&& maze[x][y] == ``1``) {` `            ``sol[x][y] = ``1``;` `            ``return` `true``;` `        ``}`   `        ``// Check if maze[x][y] is valid` `        ``if` `(isSafe(maze, x, y) == ``true``) {` `              ``// Check if the current block is already part of solution path.    ` `              ``if` `(sol[x][y] == ``1``)` `                  ``return` `false``;` `          `  `            ``// 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` `} };`   `        ``N = maze.length;` `        ``rat.solveMaze(maze);` `    ``}` `}` `// This code is contributed by Abhishek Shankhadhar`

## Python3

 `# Python3 program to solve Rat in a Maze ` `# problem using backtracking `   `# 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 feasible 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` `and` `maze[x][y]``=``=` `1``:` `        ``sol[x][y] ``=` `1` `        ``return` `True` `        `  `    ``# Check if maze[x][y] is valid` `    ``if` `isSafe(maze, x, y) ``=``=` `True``:` `        ``# Check if the current block is already part of solution path.    ` `        ``if` `sol[x][y] ``=``=` `1``:` `            ``return` `False` `          `  `        ``# 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 moving in y direction doesn't give solution then ` `        ``# Move back in x direction` `        ``if` `solveMazeUtil(maze, x ``-` `1``, y, sol) ``=``=` `True``:` `            ``return` `True` `            `  `        ``# If moving in backwards in x direction doesn't give solution ` `        ``# then Move upwards 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`

## C#

 `// C# program to solve Rat in a Maze` `// problem using backtracking ` `using` `System;`   `class` `RatMaze{`   `// Size of the maze` `static` `int` `N;`   `// 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++)` `            ``Console.Write(``" "` `+ sol[i, j] + ``" "``);` `            `  `        ``Console.WriteLine();` `    ``}` `}`   `// A utility function to check if x, y` `// is valid index for N*N maze ` `bool` `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.` `bool` `solveMaze(``int` `[,]maze)` `{` `    ``int` `[,]sol = ``new` `int``[N, N];`   `    ``if` `(solveMazeUtil(maze, 0, 0, sol) == ``false``) ` `    ``{` `        ``Console.Write(``"Solution doesn't exist"``);` `        ``return` `false``;` `    ``}`   `    ``printSolution(sol);` `    ``return` `true``;` `}`   `// A recursive utility function to solve Maze ` `// problem ` `bool` `solveMazeUtil(``int` `[,]maze, ``int` `x, ``int` `y,` `                   ``int` `[,]sol)` `{` `    `  `    ``// If (x, y is goal) return true` `    ``if` `(x == N - 1 && y == N - 1 && ` `        ``maze[x, y] == 1)` `    ``{` `        ``sol[x, y] = 1;` `        ``return` `true``;` `    ``}`   `    ``// Check if maze[x,y] is valid` `    ``if` `(isSafe(maze, x, y) == ``true``)` `    ``{` `          ``// Check if the current block is already part of solution path.    ` `          ``if` `(sol[x, y] == 1)` `              ``return` `false``;` `        `  `        ``// 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 moving in y direction doesm't give` `        ``// solution then Move backward in x direction ` `        ``if` `(solveMazeUtil(maze, x - 1, y, sol))` `            ``return` `true``;`   `        ``// If moving in backwards in x direction doesn't give ` `        ``// solution then Move upwards 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``;` `}`   `// Driver Code` `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 } };`   `    ``N = maze.GetLength(0);` `    ``rat.solveMaze(maze);` `}` `}`   `// This code is contributed by gauravrajput1`

## Javascript

 ``

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```

Complexity Analysis:

• Time Complexity: O(2^(n^2)).
The recursion can run upper-bound 2^(n^2) times.
• Space Complexity: O(n^2).
Output matrix is required so an extra space of size n*n is needed.

Below is an extended version of this problem. Count number of ways to reach destination in a Maze
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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