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Rat in a maze is also one popular problem that utilizes backtracking. I

A maze is a 2D matrix in which some cells are blocked. One of the cells is the source cell, from where we have to start. And another one of them is the destination, where we have to reach. We have to find a path from the source to the destination without moving into any of the blocked cell. A picture of an unsolved maze is shown below.

And this is its solution.

To solve this puzzle, we first start with the source cell and move in a direction where the path is not blocked. If taken path makes us reach to the destination then the puzzle is solved. Else, we come back and change our direction of the path taken. We are going to implement the same logic in our code also.

Input: maze[][] = { {0,1,0,1,1}, {0,0,0,0,0}, {1,0,1,0,1}, {0,0,1,0,0}, {1,0,0,1,0}} Output: 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1

Firstly, we will make a matrix to represent the maze, and the elements of the matrix will be either 0 or 1. 1 will represent the blocked cell and 0 will represent the cells in which we can move. The matrix for the maze shown above is:

0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0

Now, we will make one more matrix of the same dimension to store the solution. Its elements will also be either 0 or 1. 1 will represent the cells in our path and rest of the cells will be 0. The matrix representing the solution is:

1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1

Thus, we now have our matrices. Next, we will find a path from the source cell to the destination cell and the steps we will take are:

Check for the current cell, if it is the destination cell, then the puzzle is solved.

If not, then we will try to move downward and see if we can move in the downward cell or not (to move in a cell it must be vacant and not already present in the path).

If we can move there, then we will continue with the path taken to the next downward cell.

If not, we will try to move to the rightward cell. And if it is blocked or taken, we will move upward.

Similarly, if we can't move up as well, we will simply move to the left cell.

If none of the four moves (down, right, up, or left) are possible, we will simply move back and change our current path (backtracking).

Thus, the summary is that we try to move to the other cell (down, right, up, and left) from the current cell and if no movement is possible, then just come back and change the direction of the path to another cell.

printsolution → This function is just printing the solution matrix.

solvemaze → This is the actual function where we are implementing the backtracking algorithm. Firstly, we are checking of our cell is the destination cell or not if (r==SIZE-1) and (c==SIZE-1). If it is the destination cell then our puzzle is already solved. If not, then we are checking if it a valid cell to move or not. A valid cell must be in the matrix i.e., indices must between 0 to SIZE-1 r>=0 && c>=0 && r<SIZE; must not be blocked maze[r][c] == 0and must not be taken in the path solution[r][c] == 0. If it is a valid move then we are free to take it and move to the next cell. Firstly, we will try the downward cell if(solveMaze(r+1, c)). If it doesn't give us the solution then we will move to the rightward cell, and similarly to the upward and the leftward cells. If all of the cells fail to give us the solution, we will leave the cell solution[r][c] = 0 and go to some other cell.

#include <iostream> using namespace std; #define SIZE 5 //the maze problem int maze[SIZE][SIZE] = { {0,1,0,1,1}, {0,0,0,0,0}, {1,0,1,0,1}, {0,0,1,0,0}, {1,0,0,1,0} }; //matrix to store the solution int solution[SIZE][SIZE]; //function to print the solution matrix void printsolution() { int i,j; for(i=0;i<SIZE;i++) { for(j=0;j<SIZE;j++) { printf("%d\t",solution[i][j]); } printf("\n\n"); } } //function to solve the maze //using backtracking int solvemaze(int r, int c) { //if destination is reached, maze is solved //destination is the last cell(maze[SIZE-1][SIZE-1]) if((r==SIZE-1) && (c==SIZE-1) { solution[r][c] = 1; return 1; } //checking if we can visit in this cell or not //the indices of the cell must be in (0,SIZE-1) //and solution[r][c] == 0 is making sure that the cell is not already visited //maze[r][c] == 0 is making sure that the cell is not blocked if(r>=0 && c>=0 && r<SIZE && c<SIZE && solution[r][c] == 0 && maze[r][c] == 0){ //if safe to visit then visit the cell solution[r][c] = 1; //going down if(solvemaze(r+1, c)) return 1; //going right if(solvemaze(r, c+1)) return 1; //going up if(solvemaze(r-1, c)) return 1; //going left if(solvemaze(r, c-1)) return 1; //backtracking solution[r][c] = 0; return 0; } return 0; } int main() { //making all elements of the solution matrix 0 int i,j; for(i=0; i<SIZE; i++) { for(j=0; j<SIZE; j++) { solution[i][j] = 0; } } if (solvemaze(0,0)) printsolution(); else printf("No solution\n"); return 0; }

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