Modify the program from the case study to print out the path...

Modify the program from the case study to print out the path without dead ends and, possibly, with detours. For example, for an input maze $$ \begin{aligned} & 1111111 \\ & 1 \mathrm{e} 00001 \\ & 1110111 \\ & 1000001 \\ & 100 \mathrm{~m} 001 \\ & 1111111 \end{aligned} $$ the program from the case study outputs the processed maze $$ \begin{aligned} & 1111111 \\ & 1 \mathrm{e} \ldots .1 \\ & 111.111 \\ & 1 \ldots .1 \\ & 1 \ldots \mathrm{m} .1 \\ & 1111111 \\ & \text { Success } \end{aligned} $$ The modified program should, in addition, generate the path from the exit to the mouse: $$ \left[\begin{array}{ll} 1 & 1 \end{array}\right]\left[\begin{array}{ll} 1 & 2 \end{array}\right]\left[\begin{array}{ll} 1 & 3 \end{array}\right]\left[\begin{array}{ll} 2 & 3 \end{array}\right]\left[\begin{array}{ll} 3 & 3 \end{array}\right]\left[\begin{array}{ll} 3 & 4 \end{array}\right]\left[\begin{array}{ll} 3 & 5 \end{array}\right]\left[\begin{array}{ll} 4 & 5 \end{array}\right]\left[\begin{array}{ll} 4 & 4 \end{array}\right]\left[\begin{array}{ll} 4 & 3 \end{array}\right] $$ but retains a detour, $\left[\begin{array}{ll}3 & 4\end{array}\right]\left[\begin{array}{ll}3 & 5\end{array}\right]\left[\begin{array}{ll}4 & 5\end{array}\right]\left[\begin{array}{ll}4 & 4\end{array}\right]$.

Modify the program from the case study to print out the path without dead ends and, possibly, with detours. For example, for an input maze
$$
\begin{aligned}
& 1111111 \\
& 1 \mathrm{e} 00001 \\
& 1110111 \\
& 1000001 \\
& 100 \mathrm{~m} 001 \\
& 1111111
\end{aligned}
$$
the program from the case study outputs the processed maze
$$
\begin{aligned}
& 1111111 \\
& 1 \mathrm{e} \ldots .1 \\
& 111.111 \\
& 1 \ldots .1 \\
& 1 \ldots \mathrm{m} .1 \\
& 1111111 \\
& \text { Success }
\end{aligned}
$$
The modified program should, in addition, generate the path from the exit to the mouse:
$$
\left[\begin{array}{ll}
1 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & 2
\end{array}\right]\left[\begin{array}{ll}
1 & 3
\end{array}\right]\left[\begin{array}{ll}
2 & 3
\end{array}\right]\left[\begin{array}{ll}
3 & 3
\end{array}\right]\left[\begin{array}{ll}
3 & 4
\end{array}\right]\left[\begin{array}{ll}
3 & 5
\end{array}\right]\left[\begin{array}{ll}
4 & 5
\end{array}\right]\left[\begin{array}{ll}
4 & 4
\end{array}\right]\left[\begin{array}{ll}
4 & 3
\end{array}\right]
$$
but retains a detour, $\left[\begin{array}{ll}3 & 4\end{array}\right]\left[\begin{array}{ll}3 & 5\end{array}\right]\left[\begin{array}{ll}4 & 5\end{array}\right]\left[\begin{array}{ll}4 & 4\end{array}\right]$.

Key Concepts

Maze Representation

This concept involves modeling a maze as a data structure, typically a grid or array, where each cell represents a part of the maze (such as walls, open paths, start, and exit). An effective representation allows the algorithm to easily identify cell types and navigate through the maze, making it a fundamental step for any pathfinding algorithm.

Graph Traversal

Graph traversal refers to the process of visiting nodes (or cells, in the case of a maze) in a systematic way, usually using algorithms such as depth-first search (DFS) or breadth-first search (BFS). In maze solving, considering the maze as a graph where cells are nodes and adjacent open cells are connected by edges is crucial for exploring possible paths.

Pathfinding Algorithm

Pathfinding algorithms are designed to find a viable route from a starting point to a destination. In the context of maze solving, these algorithms search for a path from the maze exit to the target (or vice versa), ensuring that the algorithm can navigate through the maze while avoiding obstacles, such as walls and dead ends.

Backtracking

Backtracking is a problem-solving technique used during maze traversal where the algorithm explores paths until it reaches a dead end, then retraces its steps to try alternate routes. This approach is important for ensuring that all possible paths are considered, and it helps in discarding paths that do not lead to a solution.

Dead-End Detection

Identifying and handling dead ends is a critical concept in maze solving. A dead end is a path that leads to no solution, and the algorithm must recognize these quickly in order to avoid unnecessary exploration. This concept ensures the efficiency of the algorithm by pruning paths that do not contribute to reaching the goal.

Path Reconstruction and Detour Tracking

Once a successful path has been found through the maze, the algorithm often needs to reconstruct and output the sequence of steps from the starting point to the destination. This includes keeping track of any detours or alternative paths taken, which may be part of the solution. Detour tracking helps to illustrate not only the final effective path but also optional segments, providing a complete picture of the exploration process.

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