# Rat in a Maze

2018/02/10 22:06

Backtracking | (Rat in a Maze)

Description:

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 destination . The rat can move only in two directions : forward and down.

In the maze matrix, 0 means the block is dead end and 1 means the block can be used in the path from source and destination. Note that this is a simple version of 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 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 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.


##Backtracking Algorithm

If destination is reached
print the  solution matrix
Else
a)	Mark current cell in solution matrix as 1.
b)	Move forward 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.


##Source Code

#include <iostream>
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 << " " << sol[i][j];
}
cout << endl;
}
}

/* 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 1. 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 == N-1 && y == N-1)
{
sol[x][y] = 1;
return true;
}

if(isSafe(maze, x, y) == true)
{
// mark x, y as part of solutin path
sol[x][y] = 1;
// Move forward in x direction
if(solveMazeUtil(maze, x+1, y, sol) == true)
return true;
// Move down in y direction
if(solveMazeUtil(maze, x, y+1, sol) == true)
return true;
// If none of the above movements work than BACKTRACK:
// unmark x, y as part of solution path
sol[x][y] = 0;
return false;
}
return false;
}

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;
}


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


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