# Rat in a Maze with multiple steps or jump allowed

This is the variation of Rat in Maze

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 destination. The rat can move only in two directions: forward and down.
In the maze matrix, 0 means the block is dead end and non-zero number means the block can be used in the path from source to destination. The non-zero value of mat[i][j] indicates number of maximum jumps rat can make from cell mat[i][j].

In this variation, Rat is allowed to jump multiple steps at a time instead of 1.

Examples:

```Input : { {2, 1, 0, 0},
{3, 0, 0, 1},
{0, 1, 0, 1},
{0, 0, 0, 1}
}
Output : { {1, 0, 0, 0},
{1, 0, 0, 1},
{0, 0, 0, 1},
{0, 0, 0, 1}
}

Explanation
Rat started with M and can jump upto 2 steps right/down.
Let's try in horizontal direction -
M won't lead to solution and M is 0 which is dead end.
So, backtrack and try in down direction.
Rat jump down to M which eventually leads to solution.

Input : {
{2, 1, 0, 0},
{2, 0, 0, 1},
{0, 1, 0, 1},
{0, 0, 0, 1}
}
Output : Solution doesn't exist
```

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 non-zero
{
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/jump (for each valid steps) in horizontal direction
and recursively check if this move leads to a solution.
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 work 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``(``"\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] != 0) ` `        ``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 */` `        ``for` `(``int` `i = 1; i <= maze[x][y] && i < N; i++) { ` ` `  `            ``/* Move forward in x direction */` `            ``if` `(solveMazeUtil(maze, x + i, 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 + i, 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] = { { 2, 1, 0, 0 }, ` `                       ``{ 3, 0, 0, 1 }, ` `                       ``{ 0, 1, 0, 1 }, ` `                       ``{ 0, 0, 0, 1 } }; ` ` `  `    ``solveMaze(maze); ` `    ``return` `0; ` `} `

## Java

 `// Java program to solve Rat in a Maze problem  ` `// using backtracking ` `class` `GFG  ` `{ ` ` `  `    ``// Maze size  ` `    ``static` `int` `N = ``4``; ` ` `  `    ``/* A utility function to print solution matrix  ` `    ``sol[N][N] */` `    ``static` `void` `printSolution(``int` `sol[][])  ` `    ``{ ` `        ``for` `(``int` `i = ``0``; i < N; i++) ` `        ``{ ` `            ``for` `(``int` `j = ``0``; j < N; j++)  ` `            ``{ ` `                ``System.out.printf(``" %d "``, sol[i][j]); ` `            ``} ` `            ``System.out.printf(``"\n"``); ` `        ``} ` `    ``} ` ` `  `    ``/* A utility function to check if x, y is valid  ` `    ``index for N*N maze */` `    ``static` `boolean` `isSafe(``int` `maze[][], ``int` `x, ``int` `y)  ` `    ``{ ` `         `  `        ``// if (x, y outside maze) return false  ` `        ``if` `(x >= ``0` `&& x < N && y >= ``0` `&&  ` `             ``y < N && maze[x][y] != ``0``) ` `        ``{ ` `            ``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.*/` `    ``static` `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.printf(``"Solution doesn't exist"``); ` `            ``return` `false``; ` `        ``} ` ` `  `        ``printSolution(sol); ` `        ``return` `true``; ` `    ``} ` ` `  `    ``/* A recursive utility function to solve Maze problem */` `    ``static` `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 */` `            ``for` `(``int` `i = ``1``; i <= maze[x][y] && i < N; i++)  ` `            ``{ ` ` `  `                ``/* Move forward in x direction */` `                ``if` `(solveMazeUtil(maze, x + i, 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 + i, 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 Code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `maze[][] = {{``2``, ``1``, ``0``, ``0``}, ` `                        ``{``3``, ``0``, ``0``, ``1``}, ` `                        ``{``0``, ``1``, ``0``, ``1``}, ` `                        ``{``0``, ``0``, ``0``, ``1``}}; ` ` `  `        ``solveMaze(maze); ` `    ``} ` `} ` ` `  `// This code is contributed by Princi Singh `

## Python3

 `""" Python3 program to solve Rat in a ` `Maze problem using backtracking """` ` `  `# Maze size  ` `N ``=` `4` ` `  `""" A utility function to prsolution matrix  ` `sol """` `def` `printSolution(sol): ` `    ``for` `i ``in` `range``(N): ` `        ``for` `j ``in` `range``(N): ` `            ``print``(sol[i][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, y outside maze) return false  ` `    ``if` `(x >``=` `0` `and` `x < N ``and` `y >``=` `0` `and`  `         ``y < N ``and` `maze[x][y] !``=` `0``): ` `        ``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): ` `    ``sol ``=` `[[``0``, ``0``, ``0``, ``0``], ` `           ``[``0``, ``0``, ``0``, ``0``], ` `           ``[``0``, ``0``, ``0``, ``0``], ` `           ``[``0``, ``0``, ``0``, ``0``]] ` `    ``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 """` `        ``for` `i ``in` `range``(``1``, N): ` `            ``if` `(i <``=` `maze[x][y]): ` `                 `  `                ``""" Move forward in x direction """` `                ``if` `(solveMazeUtil(maze, x ``+` `i,  ` `                                  ``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 ``+` `i, 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 Code ` `maze ``=` `[[``2``, ``1``, ``0``, ``0``], ` `        ``[``3``, ``0``, ``0``, ``1``], ` `        ``[``0``, ``1``, ``0``, ``1``], ` `        ``[``0``, ``0``, ``0``, ``1``]] ` `solveMaze(maze)  ` ` `  `# This code is contributed by SHUBHAMSINGH10 `

## C#

 `// C# program to solve Rat in a Maze problem  ` `// using backtracking ` `using` `System; ` `     `  `class` `GFG  ` `{ ` ` `  `    ``// Maze size  ` `    ``static` `int` `N = 4; ` ` `  `    ``/* A utility function to print  ` `    ``solution matrix sol[N, N] */` `    ``static` `void` `printSolution(``int` `[,]sol)  ` `    ``{ ` `        ``for` `(``int` `i = 0; i < N; i++) ` `        ``{ ` `            ``for` `(``int` `j = 0; j < N; j++)  ` `            ``{ ` `                ``Console.Write(``" {0} "``, sol[i, j]); ` `            ``} ` `            ``Console.Write(``"\n"``); ` `        ``} ` `    ``} ` ` `  `    ``/* A utility function to check if ` `    ``x, y is valid index for N*N maze */` `    ``static` `Boolean isSafe(``int` `[,]maze, ` `                          ``int` `x, ``int` `y)  ` `    ``{ ` `         `  `        ``// if (x, y outside maze) return false  ` `        ``if` `(x >= 0 && x < N && y >= 0 &&  ` `            ``y < N && maze[x, y] != 0) ` `        ``{ ` `            ``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.*/` `    ``static` `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``)  ` `        ``{ ` `            ``Console.Write(``"Solution doesn't exist"``); ` `            ``return` `false``; ` `        ``} ` ` `  `        ``printSolution(sol); ` `        ``return` `true``; ` `    ``} ` ` `  `    ``/* A recursive utility function to solve Maze problem */` `    ``static` `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 */` `            ``for` `(``int` `i = 1; ` `                     ``i <= maze[x, y] && i < N; i++)  ` `            ``{ ` ` `  `                ``/* Move forward in x direction */` `                ``if` `(solveMazeUtil(maze, x + i,  ` `                                  ``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 + i, 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 Code ` `    ``public` `static` `void` `Main(String[] args) ` `    ``{ ` `        ``int` `[,]maze = {{2, 1, 0, 0}, ` `                       ``{3, 0, 0, 1}, ` `                       ``{0, 1, 0, 1}, ` `                       ``{0, 0, 0, 1}}; ` ` `  `        ``solveMaze(maze); ` `    ``} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

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

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